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  98. <li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code> — Generate pseudo-random numbers</a><ul>

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  100. <li><a class="reference internal" href="#functions-for-bytes">Functions for bytes</a></li>

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  201.   <section id="module-random">

  202. <span id="random-generate-pseudo-random-numbers"></span><h1><a class="reference internal" href="#module-random" title="random: Generate pseudo-random numbers with various common distributions."><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code></a> — Generate pseudo-random numbers<a class="headerlink" href="#module-random" title="Link to this heading">¶</a></h1>

  203. <p><strong>Source code:</strong> <a class="reference external" href="https://github.com/python/cpython/tree/3.12/Lib/random.py">Lib/random.py</a></p>

  204. <hr class="docutils" />

  205. <p>This module implements pseudo-random number generators for various

  206. distributions.</p>

  207. <p>For integers, there is uniform selection from a range. For sequences, there is

  208. uniform selection of a random element, a function to generate a random

  209. permutation of a list in-place, and a function for random sampling without

  210. replacement.</p>

  211. <p>On the real line, there are functions to compute uniform, normal (Gaussian),

  212. lognormal, negative exponential, gamma, and beta distributions. For generating

  213. distributions of angles, the von Mises distribution is available.</p>

  214. <p>Almost all module functions depend on the basic function <a class="reference internal" href="#random.random" title="random.random"><code class="xref py py-func docutils literal notranslate"><span class="pre">random()</span></code></a>, which

  215. generates a random float uniformly in the half-open range <code class="docutils literal notranslate"><span class="pre">0.0</span> <span class="pre">&lt;=</span> <span class="pre">X</span> <span class="pre">&lt;</span> <span class="pre">1.0</span></code>.

  216. Python uses the Mersenne Twister as the core generator.  It produces 53-bit precision

  217. floats and has a period of 2**19937-1.  The underlying implementation in C is

  218. both fast and threadsafe.  The Mersenne Twister is one of the most extensively

  219. tested random number generators in existence.  However, being completely

  220. deterministic, it is not suitable for all purposes, and is completely unsuitable

  221. for cryptographic purposes.</p>

  222. <p>The functions supplied by this module are actually bound methods of a hidden

  223. instance of the <a class="reference internal" href="#random.Random" title="random.Random"><code class="xref py py-class docutils literal notranslate"><span class="pre">random.Random</span></code></a> class.  You can instantiate your own

  224. instances of <a class="reference internal" href="#random.Random" title="random.Random"><code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code></a> to get generators that don’t share state.</p>

  225. <p>Class <a class="reference internal" href="#random.Random" title="random.Random"><code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code></a> can also be subclassed if you want to use a different

  226. basic generator of your own devising: see the documentation on that class for

  227. more details.</p>

  228. <p>The <a class="reference internal" href="#module-random" title="random: Generate pseudo-random numbers with various common distributions."><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code></a> module also provides the <a class="reference internal" href="#random.SystemRandom" title="random.SystemRandom"><code class="xref py py-class docutils literal notranslate"><span class="pre">SystemRandom</span></code></a> class which

  229. uses the system function <a class="reference internal" href="os.html#os.urandom" title="os.urandom"><code class="xref py py-func docutils literal notranslate"><span class="pre">os.urandom()</span></code></a> to generate random numbers

  230. from sources provided by the operating system.</p>

  231. <div class="admonition warning">

  232. <p class="admonition-title">Warning</p>

  233. <p>The pseudo-random generators of this module should not be used for

  234. security purposes.  For security or cryptographic uses, see the

  235. <a class="reference internal" href="secrets.html#module-secrets" title="secrets: Generate secure random numbers for managing secrets."><code class="xref py py-mod docutils literal notranslate"><span class="pre">secrets</span></code></a> module.</p>

  236. </div>

  237. <div class="admonition seealso">

  238. <p class="admonition-title">See also</p>

  239. <p>M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally

  240. equidistributed uniform pseudorandom number generator”, ACM Transactions on

  241. Modeling and Computer Simulation Vol. 8, No. 1, January pp.3–30 1998.</p>

  242. <p><a class="reference external" href="https://code.activestate.com/recipes/576707/">Complementary-Multiply-with-Carry recipe</a> for a compatible alternative

  243. random number generator with a long period and comparatively simple update

  244. operations.</p>

  245. </div>

  246. <section id="bookkeeping-functions">

  247. <h2>Bookkeeping functions<a class="headerlink" href="#bookkeeping-functions" title="Link to this heading">¶</a></h2>

  248. <dl class="py function">

  249. <dt class="sig sig-object py" id="random.seed">

  250. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">seed</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">version</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.seed" title="Link to this definition">¶</a></dt>

  251. <dd><p>Initialize the random number generator.</p>

  252. <p>If <em>a</em> is omitted or <code class="docutils literal notranslate"><span class="pre">None</span></code>, the current system time is used.  If

  253. randomness sources are provided by the operating system, they are used

  254. instead of the system time (see the <a class="reference internal" href="os.html#os.urandom" title="os.urandom"><code class="xref py py-func docutils literal notranslate"><span class="pre">os.urandom()</span></code></a> function for details

  255. on availability).</p>

  256. <p>If <em>a</em> is an int, it is used directly.</p>

  257. <p>With version 2 (the default), a <a class="reference internal" href="stdtypes.html#str" title="str"><code class="xref py py-class docutils literal notranslate"><span class="pre">str</span></code></a>, <a class="reference internal" href="stdtypes.html#bytes" title="bytes"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytes</span></code></a>, or <a class="reference internal" href="stdtypes.html#bytearray" title="bytearray"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytearray</span></code></a>

  258. object gets converted to an <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a> and all of its bits are used.</p>

  259. <p>With version 1 (provided for reproducing random sequences from older versions

  260. of Python), the algorithm for <a class="reference internal" href="stdtypes.html#str" title="str"><code class="xref py py-class docutils literal notranslate"><span class="pre">str</span></code></a> and <a class="reference internal" href="stdtypes.html#bytes" title="bytes"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytes</span></code></a> generates a

  261. narrower range of seeds.</p>

  262. <div class="versionchanged">

  263. <p><span class="versionmodified changed">Changed in version 3.2: </span>Moved to the version 2 scheme which uses all of the bits in a string seed.</p>

  264. </div>

  265. <div class="versionchanged">

  266. <p><span class="versionmodified changed">Changed in version 3.11: </span>The <em>seed</em> must be one of the following types:

  267. <code class="docutils literal notranslate"><span class="pre">None</span></code>, <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a>, <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>, <a class="reference internal" href="stdtypes.html#str" title="str"><code class="xref py py-class docutils literal notranslate"><span class="pre">str</span></code></a>,

  268. <a class="reference internal" href="stdtypes.html#bytes" title="bytes"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytes</span></code></a>, or <a class="reference internal" href="stdtypes.html#bytearray" title="bytearray"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytearray</span></code></a>.</p>

  269. </div>

  270. </dd></dl>

  271. 

  272. <dl class="py function">

  273. <dt class="sig sig-object py" id="random.getstate">

  274. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">getstate</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#random.getstate" title="Link to this definition">¶</a></dt>

  275. <dd><p>Return an object capturing the current internal state of the generator.  This

  276. object can be passed to <a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-func docutils literal notranslate"><span class="pre">setstate()</span></code></a> to restore the state.</p>

  277. </dd></dl>

  278. 

  279. <dl class="py function">

  280. <dt class="sig sig-object py" id="random.setstate">

  281. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">setstate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.setstate" title="Link to this definition">¶</a></dt>

  282. <dd><p><em>state</em> should have been obtained from a previous call to <a class="reference internal" href="#random.getstate" title="random.getstate"><code class="xref py py-func docutils literal notranslate"><span class="pre">getstate()</span></code></a>, and

  283. <a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-func docutils literal notranslate"><span class="pre">setstate()</span></code></a> restores the internal state of the generator to what it was at

  284. the time <a class="reference internal" href="#random.getstate" title="random.getstate"><code class="xref py py-func docutils literal notranslate"><span class="pre">getstate()</span></code></a> was called.</p>

  285. </dd></dl>

  286. 

  287. </section>

  288. <section id="functions-for-bytes">

  289. <h2>Functions for bytes<a class="headerlink" href="#functions-for-bytes" title="Link to this heading">¶</a></h2>

  290. <dl class="py function">

  291. <dt class="sig sig-object py" id="random.randbytes">

  292. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">randbytes</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.randbytes" title="Link to this definition">¶</a></dt>

  293. <dd><p>Generate <em>n</em> random bytes.</p>

  294. <p>This method should not be used for generating security tokens.

  295. Use <a class="reference internal" href="secrets.html#secrets.token_bytes" title="secrets.token_bytes"><code class="xref py py-func docutils literal notranslate"><span class="pre">secrets.token_bytes()</span></code></a> instead.</p>

  296. <div class="versionadded">

  297. <p><span class="versionmodified added">New in version 3.9.</span></p>

  298. </div>

  299. </dd></dl>

  300. 

  301. </section>

  302. <section id="functions-for-integers">

  303. <h2>Functions for integers<a class="headerlink" href="#functions-for-integers" title="Link to this heading">¶</a></h2>

  304. <dl class="py function">

  305. <dt class="sig sig-object py" id="random.randrange">

  306. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">randrange</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">stop</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.randrange" title="Link to this definition">¶</a></dt>

  307. <dt class="sig sig-object py">

  308. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">randrange</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">start</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">stop</span></span></em><span class="optional">[</span>, <em class="sig-param"><span class="n"><span class="pre">step</span></span></em><span class="optional">]</span><span class="sig-paren">)</span></dt>

  309. <dd><p>Return a randomly selected element from <code class="docutils literal notranslate"><span class="pre">range(start,</span> <span class="pre">stop,</span> <span class="pre">step)</span></code>.</p>

  310. <p>This is roughly equivalent to <code class="docutils literal notranslate"><span class="pre">choice(range(start,</span> <span class="pre">stop,</span> <span class="pre">step))</span></code> but

  311. supports arbitrarily large ranges and is optimized for common cases.</p>

  312. <p>The positional argument pattern matches the <a class="reference internal" href="stdtypes.html#range" title="range"><code class="xref py py-func docutils literal notranslate"><span class="pre">range()</span></code></a> function.</p>

  313. <p>Keyword arguments should not be used because they can be interpreted

  314. in unexpected ways. For example <code class="docutils literal notranslate"><span class="pre">randrange(start=100)</span></code> is interpreted

  315. as <code class="docutils literal notranslate"><span class="pre">randrange(0,</span> <span class="pre">100,</span> <span class="pre">1)</span></code>.</p>

  316. <div class="versionchanged">

  317. <p><span class="versionmodified changed">Changed in version 3.2: </span><a class="reference internal" href="#random.randrange" title="random.randrange"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randrange()</span></code></a> is more sophisticated about producing equally distributed

  318. values.  Formerly it used a style like <code class="docutils literal notranslate"><span class="pre">int(random()*n)</span></code> which could produce

  319. slightly uneven distributions.</p>

  320. </div>

  321. <div class="versionchanged">

  322. <p><span class="versionmodified changed">Changed in version 3.12: </span>Automatic conversion of non-integer types is no longer supported.

  323. Calls such as <code class="docutils literal notranslate"><span class="pre">randrange(10.0)</span></code> and <code class="docutils literal notranslate"><span class="pre">randrange(Fraction(10,</span> <span class="pre">1))</span></code>

  324. now raise a <a class="reference internal" href="exceptions.html#TypeError" title="TypeError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a>.</p>

  325. </div>

  326. </dd></dl>

  327. 

  328. <dl class="py function">

  329. <dt class="sig sig-object py" id="random.randint">

  330. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">randint</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.randint" title="Link to this definition">¶</a></dt>

  331. <dd><p>Return a random integer <em>N</em> such that <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&lt;=</span> <span class="pre">N</span> <span class="pre">&lt;=</span> <span class="pre">b</span></code>.  Alias for

  332. <code class="docutils literal notranslate"><span class="pre">randrange(a,</span> <span class="pre">b+1)</span></code>.</p>

  333. </dd></dl>

  334. 

  335. <dl class="py function">

  336. <dt class="sig sig-object py" id="random.getrandbits">

  337. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">getrandbits</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">k</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.getrandbits" title="Link to this definition">¶</a></dt>

  338. <dd><p>Returns a non-negative Python integer with <em>k</em> random bits. This method

  339. is supplied with the Mersenne Twister generator and some other generators

  340. may also provide it as an optional part of the API. When available,

  341. <a class="reference internal" href="#random.getrandbits" title="random.getrandbits"><code class="xref py py-meth docutils literal notranslate"><span class="pre">getrandbits()</span></code></a> enables <a class="reference internal" href="#random.randrange" title="random.randrange"><code class="xref py py-meth docutils literal notranslate"><span class="pre">randrange()</span></code></a> to handle arbitrarily large

  342. ranges.</p>

  343. <div class="versionchanged">

  344. <p><span class="versionmodified changed">Changed in version 3.9: </span>This method now accepts zero for <em>k</em>.</p>

  345. </div>

  346. </dd></dl>

  347. 

  348. </section>

  349. <section id="functions-for-sequences">

  350. <h2>Functions for sequences<a class="headerlink" href="#functions-for-sequences" title="Link to this heading">¶</a></h2>

  351. <dl class="py function">

  352. <dt class="sig sig-object py" id="random.choice">

  353. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">choice</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">seq</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.choice" title="Link to this definition">¶</a></dt>

  354. <dd><p>Return a random element from the non-empty sequence <em>seq</em>. If <em>seq</em> is empty,

  355. raises <a class="reference internal" href="exceptions.html#IndexError" title="IndexError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">IndexError</span></code></a>.</p>

  356. </dd></dl>

  357. 

  358. <dl class="py function">

  359. <dt class="sig sig-object py" id="random.choices">

  360. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">choices</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">population</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">weights</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">cum_weights</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.choices" title="Link to this definition">¶</a></dt>

  361. <dd><p>Return a <em>k</em> sized list of elements chosen from the <em>population</em> with replacement.

  362. If the <em>population</em> is empty, raises <a class="reference internal" href="exceptions.html#IndexError" title="IndexError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">IndexError</span></code></a>.</p>

  363. <p>If a <em>weights</em> sequence is specified, selections are made according to the

  364. relative weights.  Alternatively, if a <em>cum_weights</em> sequence is given, the

  365. selections are made according to the cumulative weights (perhaps computed

  366. using <a class="reference internal" href="itertools.html#itertools.accumulate" title="itertools.accumulate"><code class="xref py py-func docutils literal notranslate"><span class="pre">itertools.accumulate()</span></code></a>).  For example, the relative weights

  367. <code class="docutils literal notranslate"><span class="pre">[10,</span> <span class="pre">5,</span> <span class="pre">30,</span> <span class="pre">5]</span></code> are equivalent to the cumulative weights

  368. <code class="docutils literal notranslate"><span class="pre">[10,</span> <span class="pre">15,</span> <span class="pre">45,</span> <span class="pre">50]</span></code>.  Internally, the relative weights are converted to

  369. cumulative weights before making selections, so supplying the cumulative

  370. weights saves work.</p>

  371. <p>If neither <em>weights</em> nor <em>cum_weights</em> are specified, selections are made

  372. with equal probability.  If a weights sequence is supplied, it must be

  373. the same length as the <em>population</em> sequence.  It is a <a class="reference internal" href="exceptions.html#TypeError" title="TypeError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">TypeError</span></code></a>

  374. to specify both <em>weights</em> and <em>cum_weights</em>.</p>

  375. <p>The <em>weights</em> or <em>cum_weights</em> can use any numeric type that interoperates

  376. with the <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a> values returned by <a class="reference internal" href="#module-random" title="random: Generate pseudo-random numbers with various common distributions."><code class="xref py py-func docutils literal notranslate"><span class="pre">random()</span></code></a> (that includes

  377. integers, floats, and fractions but excludes decimals).  Weights are assumed

  378. to be non-negative and finite.  A <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> is raised if all

  379. weights are zero.</p>

  380. <p>For a given seed, the <a class="reference internal" href="#random.choices" title="random.choices"><code class="xref py py-func docutils literal notranslate"><span class="pre">choices()</span></code></a> function with equal weighting

  381. typically produces a different sequence than repeated calls to

  382. <a class="reference internal" href="#random.choice" title="random.choice"><code class="xref py py-func docutils literal notranslate"><span class="pre">choice()</span></code></a>.  The algorithm used by <a class="reference internal" href="#random.choices" title="random.choices"><code class="xref py py-func docutils literal notranslate"><span class="pre">choices()</span></code></a> uses floating

  383. point arithmetic for internal consistency and speed.  The algorithm used

  384. by <a class="reference internal" href="#random.choice" title="random.choice"><code class="xref py py-func docutils literal notranslate"><span class="pre">choice()</span></code></a> defaults to integer arithmetic with repeated selections

  385. to avoid small biases from round-off error.</p>

  386. <div class="versionadded">

  387. <p><span class="versionmodified added">New in version 3.6.</span></p>

  388. </div>

  389. <div class="versionchanged">

  390. <p><span class="versionmodified changed">Changed in version 3.9: </span>Raises a <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a> if all weights are zero.</p>

  391. </div>

  392. </dd></dl>

  393. 

  394. <dl class="py function">

  395. <dt class="sig sig-object py" id="random.shuffle">

  396. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">shuffle</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">x</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.shuffle" title="Link to this definition">¶</a></dt>

  397. <dd><p>Shuffle the sequence <em>x</em> in place.</p>

  398. <p>To shuffle an immutable sequence and return a new shuffled list, use

  399. <code class="docutils literal notranslate"><span class="pre">sample(x,</span> <span class="pre">k=len(x))</span></code> instead.</p>

  400. <p>Note that even for small <code class="docutils literal notranslate"><span class="pre">len(x)</span></code>, the total number of permutations of <em>x</em>

  401. can quickly grow larger than the period of most random number generators.

  402. This implies that most permutations of a long sequence can never be

  403. generated.  For example, a sequence of length 2080 is the largest that

  404. can fit within the period of the Mersenne Twister random number generator.</p>

  405. <div class="versionchanged">

  406. <p><span class="versionmodified changed">Changed in version 3.11: </span>Removed the optional parameter <em>random</em>.</p>

  407. </div>

  408. </dd></dl>

  409. 

  410. <dl class="py function">

  411. <dt class="sig sig-object py" id="random.sample">

  412. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">sample</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">population</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">k</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">counts</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.sample" title="Link to this definition">¶</a></dt>

  413. <dd><p>Return a <em>k</em> length list of unique elements chosen from the population

  414. sequence.  Used for random sampling without replacement.</p>

  415. <p>Returns a new list containing elements from the population while leaving the

  416. original population unchanged.  The resulting list is in selection order so that

  417. all sub-slices will also be valid random samples.  This allows raffle winners

  418. (the sample) to be partitioned into grand prize and second place winners (the

  419. subslices).</p>

  420. <p>Members of the population need not be <a class="reference internal" href="../glossary.html#term-hashable"><span class="xref std std-term">hashable</span></a> or unique.  If the population

  421. contains repeats, then each occurrence is a possible selection in the sample.</p>

  422. <p>Repeated elements can be specified one at a time or with the optional

  423. keyword-only <em>counts</em> parameter.  For example, <code class="docutils literal notranslate"><span class="pre">sample(['red',</span> <span class="pre">'blue'],</span>

  424. <span class="pre">counts=[4,</span> <span class="pre">2],</span> <span class="pre">k=5)</span></code> is equivalent to <code class="docutils literal notranslate"><span class="pre">sample(['red',</span> <span class="pre">'red',</span> <span class="pre">'red',</span> <span class="pre">'red',</span>

  425. <span class="pre">'blue',</span> <span class="pre">'blue'],</span> <span class="pre">k=5)</span></code>.</p>

  426. <p>To choose a sample from a range of integers, use a <a class="reference internal" href="stdtypes.html#range" title="range"><code class="xref py py-func docutils literal notranslate"><span class="pre">range()</span></code></a> object as an

  427. argument.  This is especially fast and space efficient for sampling from a large

  428. population:  <code class="docutils literal notranslate"><span class="pre">sample(range(10000000),</span> <span class="pre">k=60)</span></code>.</p>

  429. <p>If the sample size is larger than the population size, a <a class="reference internal" href="exceptions.html#ValueError" title="ValueError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">ValueError</span></code></a>

  430. is raised.</p>

  431. <div class="versionchanged">

  432. <p><span class="versionmodified changed">Changed in version 3.9: </span>Added the <em>counts</em> parameter.</p>

  433. </div>

  434. <div class="versionchanged">

  435. <p><span class="versionmodified changed">Changed in version 3.11: </span>The <em>population</em> must be a sequence.  Automatic conversion of sets

  436. to lists is no longer supported.</p>

  437. </div>

  438. </dd></dl>

  439. 

  440. </section>

  441. <section id="discrete-distributions">

  442. <h2>Discrete distributions<a class="headerlink" href="#discrete-distributions" title="Link to this heading">¶</a></h2>

  443. <p>The following function generates a discrete distribution.</p>

  444. <dl class="py function">

  445. <dt class="sig sig-object py" id="random.binomialvariate">

  446. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">binomialvariate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">n</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">p</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.5</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.binomialvariate" title="Link to this definition">¶</a></dt>

  447. <dd><p><a class="reference external" href="https://mathworld.wolfram.com/BinomialDistribution.html">Binomial distribution</a>.

  448. Return the number of successes for <em>n</em> independent trials with the

  449. probability of success in each trial being <em>p</em>:</p>

  450. <p>Mathematically equivalent to:</p>

  451. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="nb">sum</span><span class="p">(</span><span class="n">random</span><span class="p">()</span> <span class="o">&lt;</span> <span class="n">p</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>

  452. </pre></div>

  453. </div>

  454. <p>The number of trials <em>n</em> should be a non-negative integer.

  455. The probability of success <em>p</em> should be between <code class="docutils literal notranslate"><span class="pre">0.0</span> <span class="pre">&lt;=</span> <span class="pre">p</span> <span class="pre">&lt;=</span> <span class="pre">1.0</span></code>.

  456. The result is an integer in the range <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre">&lt;=</span> <span class="pre">X</span> <span class="pre">&lt;=</span> <span class="pre">n</span></code>.</p>

  457. <div class="versionadded">

  458. <p><span class="versionmodified added">New in version 3.12.</span></p>

  459. </div>

  460. </dd></dl>

  461. 

  462. </section>

  463. <section id="real-valued-distributions">

  464. <span id="id1"></span><h2>Real-valued distributions<a class="headerlink" href="#real-valued-distributions" title="Link to this heading">¶</a></h2>

  465. <p>The following functions generate specific real-valued distributions. Function

  466. parameters are named after the corresponding variables in the distribution’s

  467. equation, as used in common mathematical practice; most of these equations can

  468. be found in any statistics text.</p>

  469. <dl class="py function">

  470. <dt class="sig sig-object py" id="random.random">

  471. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">random</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#random.random" title="Link to this definition">¶</a></dt>

  472. <dd><p>Return the next random floating point number in the range <code class="docutils literal notranslate"><span class="pre">0.0</span> <span class="pre">&lt;=</span> <span class="pre">X</span> <span class="pre">&lt;</span> <span class="pre">1.0</span></code></p>

  473. </dd></dl>

  474. 

  475. <dl class="py function">

  476. <dt class="sig sig-object py" id="random.uniform">

  477. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">uniform</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">b</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.uniform" title="Link to this definition">¶</a></dt>

  478. <dd><p>Return a random floating point number <em>N</em> such that <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&lt;=</span> <span class="pre">N</span> <span class="pre">&lt;=</span> <span class="pre">b</span></code> for

  479. <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&lt;=</span> <span class="pre">b</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">&lt;=</span> <span class="pre">N</span> <span class="pre">&lt;=</span> <span class="pre">a</span></code> for <code class="docutils literal notranslate"><span class="pre">b</span> <span class="pre">&lt;</span> <span class="pre">a</span></code>.</p>

  480. <p>The end-point value <code class="docutils literal notranslate"><span class="pre">b</span></code> may or may not be included in the range

  481. depending on floating-point rounding in the expression

  482. <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">+</span> <span class="pre">(b-a)</span> <span class="pre">*</span> <span class="pre">random()</span></code>.</p>

  483. </dd></dl>

  484. 

  485. <dl class="py function">

  486. <dt class="sig sig-object py" id="random.triangular">

  487. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">triangular</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">low</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">high</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">mode</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.triangular" title="Link to this definition">¶</a></dt>

  488. <dd><p>Return a random floating point number <em>N</em> such that <code class="docutils literal notranslate"><span class="pre">low</span> <span class="pre">&lt;=</span> <span class="pre">N</span> <span class="pre">&lt;=</span> <span class="pre">high</span></code> and

  489. with the specified <em>mode</em> between those bounds.  The <em>low</em> and <em>high</em> bounds

  490. default to zero and one.  The <em>mode</em> argument defaults to the midpoint

  491. between the bounds, giving a symmetric distribution.</p>

  492. </dd></dl>

  493. 

  494. <dl class="py function">

  495. <dt class="sig sig-object py" id="random.betavariate">

  496. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">betavariate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">beta</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.betavariate" title="Link to this definition">¶</a></dt>

  497. <dd><p>Beta distribution.  Conditions on the parameters are <code class="docutils literal notranslate"><span class="pre">alpha</span> <span class="pre">&gt;</span> <span class="pre">0</span></code> and

  498. <code class="docutils literal notranslate"><span class="pre">beta</span> <span class="pre">&gt;</span> <span class="pre">0</span></code>. Returned values range between 0 and 1.</p>

  499. </dd></dl>

  500. 

  501. <dl class="py function">

  502. <dt class="sig sig-object py" id="random.expovariate">

  503. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">expovariate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">lambd</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.expovariate" title="Link to this definition">¶</a></dt>

  504. <dd><p>Exponential distribution.  <em>lambd</em> is 1.0 divided by the desired

  505. mean.  It should be nonzero.  (The parameter would be called

  506. “lambda”, but that is a reserved word in Python.)  Returned values

  507. range from 0 to positive infinity if <em>lambd</em> is positive, and from

  508. negative infinity to 0 if <em>lambd</em> is negative.</p>

  509. <div class="versionchanged">

  510. <p><span class="versionmodified changed">Changed in version 3.12: </span>Added the default value for <code class="docutils literal notranslate"><span class="pre">lambd</span></code>.</p>

  511. </div>

  512. </dd></dl>

  513. 

  514. <dl class="py function">

  515. <dt class="sig sig-object py" id="random.gammavariate">

  516. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">gammavariate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">beta</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.gammavariate" title="Link to this definition">¶</a></dt>

  517. <dd><p>Gamma distribution.  (<em>Not</em> the gamma function!)  The shape and

  518. scale parameters, <em>alpha</em> and <em>beta</em>, must have positive values.

  519. (Calling conventions vary and some sources define ‘beta’

  520. as the inverse of the scale).</p>

  521. <p>The probability distribution function is:</p>

  522. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span>          <span class="n">x</span> <span class="o">**</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span> <span class="o">/</span> <span class="n">beta</span><span class="p">)</span>

  523. <span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">=</span>  <span class="o">--------------------------------------</span>

  524.             <span class="n">math</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">alpha</span><span class="p">)</span> <span class="o">*</span> <span class="n">beta</span> <span class="o">**</span> <span class="n">alpha</span>

  525. </pre></div>

  526. </div>

  527. </dd></dl>

  528. 

  529. <dl class="py function">

  530. <dt class="sig sig-object py" id="random.gauss">

  531. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">gauss</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mu</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.gauss" title="Link to this definition">¶</a></dt>

  532. <dd><p>Normal distribution, also called the Gaussian distribution.

  533. <em>mu</em> is the mean,

  534. and <em>sigma</em> is the standard deviation.  This is slightly faster than

  535. the <a class="reference internal" href="#random.normalvariate" title="random.normalvariate"><code class="xref py py-func docutils literal notranslate"><span class="pre">normalvariate()</span></code></a> function defined below.</p>

  536. <p>Multithreading note:  When two threads call this function

  537. simultaneously, it is possible that they will receive the

  538. same return value.  This can be avoided in three ways.

  539. 1) Have each thread use a different instance of the random

  540. number generator. 2) Put locks around all calls. 3) Use the

  541. slower, but thread-safe <a class="reference internal" href="#random.normalvariate" title="random.normalvariate"><code class="xref py py-func docutils literal notranslate"><span class="pre">normalvariate()</span></code></a> function instead.</p>

  542. <div class="versionchanged">

  543. <p><span class="versionmodified changed">Changed in version 3.11: </span><em>mu</em> and <em>sigma</em> now have default arguments.</p>

  544. </div>

  545. </dd></dl>

  546. 

  547. <dl class="py function">

  548. <dt class="sig sig-object py" id="random.lognormvariate">

  549. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">lognormvariate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mu</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.lognormvariate" title="Link to this definition">¶</a></dt>

  550. <dd><p>Log normal distribution.  If you take the natural logarithm of this

  551. distribution, you’ll get a normal distribution with mean <em>mu</em> and standard

  552. deviation <em>sigma</em>.  <em>mu</em> can have any value, and <em>sigma</em> must be greater than

  553. zero.</p>

  554. </dd></dl>

  555. 

  556. <dl class="py function">

  557. <dt class="sig sig-object py" id="random.normalvariate">

  558. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">normalvariate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mu</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">0.0</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">sigma</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">1.0</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.normalvariate" title="Link to this definition">¶</a></dt>

  559. <dd><p>Normal distribution.  <em>mu</em> is the mean, and <em>sigma</em> is the standard deviation.</p>

  560. <div class="versionchanged">

  561. <p><span class="versionmodified changed">Changed in version 3.11: </span><em>mu</em> and <em>sigma</em> now have default arguments.</p>

  562. </div>

  563. </dd></dl>

  564. 

  565. <dl class="py function">

  566. <dt class="sig sig-object py" id="random.vonmisesvariate">

  567. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">vonmisesvariate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">mu</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">kappa</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.vonmisesvariate" title="Link to this definition">¶</a></dt>

  568. <dd><p><em>mu</em> is the mean angle, expressed in radians between 0 and 2*<em>pi</em>, and <em>kappa</em>

  569. is the concentration parameter, which must be greater than or equal to zero.  If

  570. <em>kappa</em> is equal to zero, this distribution reduces to a uniform random angle

  571. over the range 0 to 2*<em>pi</em>.</p>

  572. </dd></dl>

  573. 

  574. <dl class="py function">

  575. <dt class="sig sig-object py" id="random.paretovariate">

  576. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">paretovariate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.paretovariate" title="Link to this definition">¶</a></dt>

  577. <dd><p>Pareto distribution.  <em>alpha</em> is the shape parameter.</p>

  578. </dd></dl>

  579. 

  580. <dl class="py function">

  581. <dt class="sig sig-object py" id="random.weibullvariate">

  582. <span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">weibullvariate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">beta</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.weibullvariate" title="Link to this definition">¶</a></dt>

  583. <dd><p>Weibull distribution.  <em>alpha</em> is the scale parameter and <em>beta</em> is the shape

  584. parameter.</p>

  585. </dd></dl>

  586. 

  587. </section>

  588. <section id="alternative-generator">

  589. <h2>Alternative Generator<a class="headerlink" href="#alternative-generator" title="Link to this heading">¶</a></h2>

  590. <dl class="py class">

  591. <dt class="sig sig-object py" id="random.Random">

  592. <em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">Random</span></span><span class="sig-paren">(</span><span class="optional">[</span><em class="sig-param"><span class="n"><span class="pre">seed</span></span></em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#random.Random" title="Link to this definition">¶</a></dt>

  593. <dd><p>Class that implements the default pseudo-random number generator used by the

  594. <a class="reference internal" href="#module-random" title="random: Generate pseudo-random numbers with various common distributions."><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code></a> module.</p>

  595. <div class="versionchanged">

  596. <p><span class="versionmodified changed">Changed in version 3.11: </span>Formerly the <em>seed</em> could be any hashable object.  Now it is limited to:

  597. <code class="docutils literal notranslate"><span class="pre">None</span></code>, <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate"><span class="pre">int</span></code></a>, <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate"><span class="pre">float</span></code></a>, <a class="reference internal" href="stdtypes.html#str" title="str"><code class="xref py py-class docutils literal notranslate"><span class="pre">str</span></code></a>,

  598. <a class="reference internal" href="stdtypes.html#bytes" title="bytes"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytes</span></code></a>, or <a class="reference internal" href="stdtypes.html#bytearray" title="bytearray"><code class="xref py py-class docutils literal notranslate"><span class="pre">bytearray</span></code></a>.</p>

  599. </div>

  600. <p>Subclasses of <code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code> should override the following methods if they

  601. wish to make use of a different basic generator:</p>

  602. <dl class="py method">

  603. <dt class="sig sig-object py" id="random.Random.seed">

  604. <span class="sig-name descname"><span class="pre">seed</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">a</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">version</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">2</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.Random.seed" title="Link to this definition">¶</a></dt>

  605. <dd><p>Override this method in subclasses to customise the <a class="reference internal" href="#random.seed" title="random.seed"><code class="xref py py-meth docutils literal notranslate"><span class="pre">seed()</span></code></a>

  606. behaviour of <code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code> instances.</p>

  607. </dd></dl>

  608. 

  609. <dl class="py method">

  610. <dt class="sig sig-object py" id="random.Random.getstate">

  611. <span class="sig-name descname"><span class="pre">getstate</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#random.Random.getstate" title="Link to this definition">¶</a></dt>

  612. <dd><p>Override this method in subclasses to customise the <a class="reference internal" href="#random.getstate" title="random.getstate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">getstate()</span></code></a>

  613. behaviour of <code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code> instances.</p>

  614. </dd></dl>

  615. 

  616. <dl class="py method">

  617. <dt class="sig sig-object py" id="random.Random.setstate">

  618. <span class="sig-name descname"><span class="pre">setstate</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">state</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.Random.setstate" title="Link to this definition">¶</a></dt>

  619. <dd><p>Override this method in subclasses to customise the <a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">setstate()</span></code></a>

  620. behaviour of <code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code> instances.</p>

  621. </dd></dl>

  622. 

  623. <dl class="py method">

  624. <dt class="sig sig-object py" id="random.Random.random">

  625. <span class="sig-name descname"><span class="pre">random</span></span><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="headerlink" href="#random.Random.random" title="Link to this definition">¶</a></dt>

  626. <dd><p>Override this method in subclasses to customise the <a class="reference internal" href="#random.random" title="random.random"><code class="xref py py-meth docutils literal notranslate"><span class="pre">random()</span></code></a>

  627. behaviour of <code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code> instances.</p>

  628. </dd></dl>

  629. 

  630. <p>Optionally, a custom generator subclass can also supply the following method:</p>

  631. <dl class="py method">

  632. <dt class="sig sig-object py" id="random.Random.getrandbits">

  633. <span class="sig-name descname"><span class="pre">getrandbits</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">k</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#random.Random.getrandbits" title="Link to this definition">¶</a></dt>

  634. <dd><p>Override this method in subclasses to customise the

  635. <a class="reference internal" href="#random.getrandbits" title="random.getrandbits"><code class="xref py py-meth docutils literal notranslate"><span class="pre">getrandbits()</span></code></a> behaviour of <code class="xref py py-class docutils literal notranslate"><span class="pre">Random</span></code> instances.</p>

  636. </dd></dl>

  637. 

  638. </dd></dl>

  639. 

  640. <dl class="py class">

  641. <dt class="sig sig-object py" id="random.SystemRandom">

  642. <em class="property"><span class="pre">class</span><span class="w"> </span></em><span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">SystemRandom</span></span><span class="sig-paren">(</span><span class="optional">[</span><em class="sig-param"><span class="n"><span class="pre">seed</span></span></em><span class="optional">]</span><span class="sig-paren">)</span><a class="headerlink" href="#random.SystemRandom" title="Link to this definition">¶</a></dt>

  643. <dd><p>Class that uses the <a class="reference internal" href="os.html#os.urandom" title="os.urandom"><code class="xref py py-func docutils literal notranslate"><span class="pre">os.urandom()</span></code></a> function for generating random numbers

  644. from sources provided by the operating system. Not available on all systems.

  645. Does not rely on software state, and sequences are not reproducible. Accordingly,

  646. the <a class="reference internal" href="#random.seed" title="random.seed"><code class="xref py py-meth docutils literal notranslate"><span class="pre">seed()</span></code></a> method has no effect and is ignored.

  647. The <a class="reference internal" href="#random.getstate" title="random.getstate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">getstate()</span></code></a> and <a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-meth docutils literal notranslate"><span class="pre">setstate()</span></code></a> methods raise

  648. <a class="reference internal" href="exceptions.html#NotImplementedError" title="NotImplementedError"><code class="xref py py-exc docutils literal notranslate"><span class="pre">NotImplementedError</span></code></a> if called.</p>

  649. </dd></dl>

  650. 

  651. </section>

  652. <section id="notes-on-reproducibility">

  653. <h2>Notes on Reproducibility<a class="headerlink" href="#notes-on-reproducibility" title="Link to this heading">¶</a></h2>

  654. <p>Sometimes it is useful to be able to reproduce the sequences given by a

  655. pseudo-random number generator.  By reusing a seed value, the same sequence should be

  656. reproducible from run to run as long as multiple threads are not running.</p>

  657. <p>Most of the random module’s algorithms and seeding functions are subject to

  658. change across Python versions, but two aspects are guaranteed not to change:</p>

  659. <ul class="simple">

  660. <li><p>If a new seeding method is added, then a backward compatible seeder will be

  661. offered.</p></li>

  662. <li><p>The generator’s <a class="reference internal" href="#random.Random.random" title="random.Random.random"><code class="xref py py-meth docutils literal notranslate"><span class="pre">random()</span></code></a> method will continue to produce the same

  663. sequence when the compatible seeder is given the same seed.</p></li>

  664. </ul>

  665. </section>

  666. <section id="examples">

  667. <span id="random-examples"></span><h2>Examples<a class="headerlink" href="#examples" title="Link to this heading">¶</a></h2>

  668. <p>Basic examples:</p>

  669. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">random</span><span class="p">()</span>                          <span class="c1"># Random float:  0.0 &lt;= x &lt; 1.0</span>

  670. <span class="go">0.37444887175646646</span>

  671. 

  672. <span class="gp">&gt;&gt;&gt; </span><span class="n">uniform</span><span class="p">(</span><span class="mf">2.5</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">)</span>                <span class="c1"># Random float:  2.5 &lt;= x &lt;= 10.0</span>

  673. <span class="go">3.1800146073117523</span>

  674. 

  675. <span class="gp">&gt;&gt;&gt; </span><span class="n">expovariate</span><span class="p">(</span><span class="mi">1</span> <span class="o">/</span> <span class="mi">5</span><span class="p">)</span>                <span class="c1"># Interval between arrivals averaging 5 seconds</span>

  676. <span class="go">5.148957571865031</span>

  677. 

  678. <span class="gp">&gt;&gt;&gt; </span><span class="n">randrange</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>                     <span class="c1"># Integer from 0 to 9 inclusive</span>

  679. <span class="go">7</span>

  680. 

  681. <span class="gp">&gt;&gt;&gt; </span><span class="n">randrange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">101</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>              <span class="c1"># Even integer from 0 to 100 inclusive</span>

  682. <span class="go">26</span>

  683. 

  684. <span class="gp">&gt;&gt;&gt; </span><span class="n">choice</span><span class="p">([</span><span class="s1">&#39;win&#39;</span><span class="p">,</span> <span class="s1">&#39;lose&#39;</span><span class="p">,</span> <span class="s1">&#39;draw&#39;</span><span class="p">])</span>   <span class="c1"># Single random element from a sequence</span>

  685. <span class="go">&#39;draw&#39;</span>

  686. 

  687. <span class="gp">&gt;&gt;&gt; </span><span class="n">deck</span> <span class="o">=</span> <span class="s1">&#39;ace two three four&#39;</span><span class="o">.</span><span class="n">split</span><span class="p">()</span>

  688. <span class="gp">&gt;&gt;&gt; </span><span class="n">shuffle</span><span class="p">(</span><span class="n">deck</span><span class="p">)</span>                     <span class="c1"># Shuffle a list</span>

  689. <span class="gp">&gt;&gt;&gt; </span><span class="n">deck</span>

  690. <span class="go">[&#39;four&#39;, &#39;two&#39;, &#39;ace&#39;, &#39;three&#39;]</span>

  691. 

  692. <span class="gp">&gt;&gt;&gt; </span><span class="n">sample</span><span class="p">([</span><span class="mi">10</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">30</span><span class="p">,</span> <span class="mi">40</span><span class="p">,</span> <span class="mi">50</span><span class="p">],</span> <span class="n">k</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span> <span class="c1"># Four samples without replacement</span>

  693. <span class="go">[40, 10, 50, 30]</span>

  694. </pre></div>

  695. </div>

  696. <p>Simulations:</p>

  697. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="c1"># Six roulette wheel spins (weighted sampling with replacement)</span>

  698. <span class="gp">&gt;&gt;&gt; </span><span class="n">choices</span><span class="p">([</span><span class="s1">&#39;red&#39;</span><span class="p">,</span> <span class="s1">&#39;black&#39;</span><span class="p">,</span> <span class="s1">&#39;green&#39;</span><span class="p">],</span> <span class="p">[</span><span class="mi">18</span><span class="p">,</span> <span class="mi">18</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="n">k</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span>

  699. <span class="go">[&#39;red&#39;, &#39;green&#39;, &#39;black&#39;, &#39;black&#39;, &#39;red&#39;, &#39;black&#39;]</span>

  700. 

  701. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Deal 20 cards without replacement from a deck</span>

  702. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># of 52 playing cards, and determine the proportion of cards</span>

  703. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># with a ten-value:  ten, jack, queen, or king.</span>

  704. <span class="gp">&gt;&gt;&gt; </span><span class="n">deal</span> <span class="o">=</span> <span class="n">sample</span><span class="p">([</span><span class="s1">&#39;tens&#39;</span><span class="p">,</span> <span class="s1">&#39;low cards&#39;</span><span class="p">],</span> <span class="n">counts</span><span class="o">=</span><span class="p">[</span><span class="mi">16</span><span class="p">,</span> <span class="mi">36</span><span class="p">],</span> <span class="n">k</span><span class="o">=</span><span class="mi">20</span><span class="p">)</span>

  705. <span class="gp">&gt;&gt;&gt; </span><span class="n">deal</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="s1">&#39;tens&#39;</span><span class="p">)</span> <span class="o">/</span> <span class="mi">20</span>

  706. <span class="go">0.15</span>

  707. 

  708. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Estimate the probability of getting 5 or more heads from 7 spins</span>

  709. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># of a biased coin that settles on heads 60% of the time.</span>

  710. <span class="gp">&gt;&gt;&gt; </span><span class="nb">sum</span><span class="p">(</span><span class="n">binomialvariate</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">7</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mf">0.6</span><span class="p">)</span> <span class="o">&gt;=</span> <span class="mi">5</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10_000</span><span class="p">))</span> <span class="o">/</span> <span class="mi">10_000</span>

  711. <span class="go">0.4169</span>

  712. 

  713. <span class="gp">&gt;&gt;&gt; </span><span class="c1"># Probability of the median of 5 samples being in middle two quartiles</span>

  714. <span class="gp">&gt;&gt;&gt; </span><span class="k">def</span> <span class="nf">trial</span><span class="p">():</span>

  715. <span class="gp">... </span>    <span class="k">return</span> <span class="mi">2_500</span> <span class="o">&lt;=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">choices</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">10_000</span><span class="p">),</span> <span class="n">k</span><span class="o">=</span><span class="mi">5</span><span class="p">))[</span><span class="mi">2</span><span class="p">]</span> <span class="o">&lt;</span> <span class="mi">7_500</span>

  716. <span class="gp">...</span>

  717. <span class="gp">&gt;&gt;&gt; </span><span class="nb">sum</span><span class="p">(</span><span class="n">trial</span><span class="p">()</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10_000</span><span class="p">))</span> <span class="o">/</span> <span class="mi">10_000</span>

  718. <span class="go">0.7958</span>

  719. </pre></div>

  720. </div>

  721. <p>Example of <a class="reference external" href="https://en.wikipedia.org/wiki/Bootstrapping_(statistics)">statistical bootstrapping</a> using resampling

  722. with replacement to estimate a confidence interval for the mean of a sample:</p>

  723. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># https://www.thoughtco.com/example-of-bootstrapping-3126155</span>

  724. <span class="kn">from</span> <span class="nn">statistics</span> <span class="kn">import</span> <span class="n">fmean</span> <span class="k">as</span> <span class="n">mean</span>

  725. <span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">choices</span>

  726. 

  727. <span class="n">data</span> <span class="o">=</span> <span class="p">[</span><span class="mi">41</span><span class="p">,</span> <span class="mi">50</span><span class="p">,</span> <span class="mi">29</span><span class="p">,</span> <span class="mi">37</span><span class="p">,</span> <span class="mi">81</span><span class="p">,</span> <span class="mi">30</span><span class="p">,</span> <span class="mi">73</span><span class="p">,</span> <span class="mi">63</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">35</span><span class="p">,</span> <span class="mi">68</span><span class="p">,</span> <span class="mi">22</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="mi">31</span><span class="p">,</span> <span class="mi">95</span><span class="p">]</span>

  728. <span class="n">means</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">mean</span><span class="p">(</span><span class="n">choices</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">data</span><span class="p">)))</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">100</span><span class="p">))</span>

  729. <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;The sample mean of </span><span class="si">{</span><span class="n">mean</span><span class="p">(</span><span class="n">data</span><span class="p">)</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1"> has a 90% confidence &#39;</span>

  730.       <span class="sa">f</span><span class="s1">&#39;interval from </span><span class="si">{</span><span class="n">means</span><span class="p">[</span><span class="mi">5</span><span class="p">]</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1"> to </span><span class="si">{</span><span class="n">means</span><span class="p">[</span><span class="mi">94</span><span class="p">]</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>

  731. </pre></div>

  732. </div>

  733. <p>Example of a <a class="reference external" href="https://en.wikipedia.org/wiki/Resampling_(statistics)#Permutation_tests">resampling permutation test</a>

  734. to determine the statistical significance or <a class="reference external" href="https://en.wikipedia.org/wiki/P-value">p-value</a> of an observed difference

  735. between the effects of a drug versus a placebo:</p>

  736. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Example from &quot;Statistics is Easy&quot; by Dennis Shasha and Manda Wilson</span>

  737. <span class="kn">from</span> <span class="nn">statistics</span> <span class="kn">import</span> <span class="n">fmean</span> <span class="k">as</span> <span class="n">mean</span>

  738. <span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">shuffle</span>

  739. 

  740. <span class="n">drug</span> <span class="o">=</span> <span class="p">[</span><span class="mi">54</span><span class="p">,</span> <span class="mi">73</span><span class="p">,</span> <span class="mi">53</span><span class="p">,</span> <span class="mi">70</span><span class="p">,</span> <span class="mi">73</span><span class="p">,</span> <span class="mi">68</span><span class="p">,</span> <span class="mi">52</span><span class="p">,</span> <span class="mi">65</span><span class="p">,</span> <span class="mi">65</span><span class="p">]</span>

  741. <span class="n">placebo</span> <span class="o">=</span> <span class="p">[</span><span class="mi">54</span><span class="p">,</span> <span class="mi">51</span><span class="p">,</span> <span class="mi">58</span><span class="p">,</span> <span class="mi">44</span><span class="p">,</span> <span class="mi">55</span><span class="p">,</span> <span class="mi">52</span><span class="p">,</span> <span class="mi">42</span><span class="p">,</span> <span class="mi">47</span><span class="p">,</span> <span class="mi">58</span><span class="p">,</span> <span class="mi">46</span><span class="p">]</span>

  742. <span class="n">observed_diff</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">drug</span><span class="p">)</span> <span class="o">-</span> <span class="n">mean</span><span class="p">(</span><span class="n">placebo</span><span class="p">)</span>

  743. 

  744. <span class="n">n</span> <span class="o">=</span> <span class="mi">10_000</span>

  745. <span class="n">count</span> <span class="o">=</span> <span class="mi">0</span>

  746. <span class="n">combined</span> <span class="o">=</span> <span class="n">drug</span> <span class="o">+</span> <span class="n">placebo</span>

  747. <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>

  748.     <span class="n">shuffle</span><span class="p">(</span><span class="n">combined</span><span class="p">)</span>

  749.     <span class="n">new_diff</span> <span class="o">=</span> <span class="n">mean</span><span class="p">(</span><span class="n">combined</span><span class="p">[:</span><span class="nb">len</span><span class="p">(</span><span class="n">drug</span><span class="p">)])</span> <span class="o">-</span> <span class="n">mean</span><span class="p">(</span><span class="n">combined</span><span class="p">[</span><span class="nb">len</span><span class="p">(</span><span class="n">drug</span><span class="p">):])</span>

  750.     <span class="n">count</span> <span class="o">+=</span> <span class="p">(</span><span class="n">new_diff</span> <span class="o">&gt;=</span> <span class="n">observed_diff</span><span class="p">)</span>

  751. 

  752. <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;</span><span class="si">{</span><span class="n">n</span><span class="si">}</span><span class="s1"> label reshufflings produced only </span><span class="si">{</span><span class="n">count</span><span class="si">}</span><span class="s1"> instances with a difference&#39;</span><span class="p">)</span>

  753. <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;at least as extreme as the observed difference of </span><span class="si">{</span><span class="n">observed_diff</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">.&#39;</span><span class="p">)</span>

  754. <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;The one-sided p-value of </span><span class="si">{</span><span class="n">count</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="n">n</span><span class="si">:</span><span class="s1">.4f</span><span class="si">}</span><span class="s1"> leads us to reject the null&#39;</span><span class="p">)</span>

  755. <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;hypothesis that there is no difference between the drug and the placebo.&#39;</span><span class="p">)</span>

  756. </pre></div>

  757. </div>

  758. <p>Simulation of arrival times and service deliveries for a multiserver queue:</p>

  759. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">heapq</span> <span class="kn">import</span> <span class="n">heapify</span><span class="p">,</span> <span class="n">heapreplace</span>

  760. <span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">expovariate</span><span class="p">,</span> <span class="n">gauss</span>

  761. <span class="kn">from</span> <span class="nn">statistics</span> <span class="kn">import</span> <span class="n">mean</span><span class="p">,</span> <span class="n">quantiles</span>

  762. 

  763. <span class="n">average_arrival_interval</span> <span class="o">=</span> <span class="mf">5.6</span>

  764. <span class="n">average_service_time</span> <span class="o">=</span> <span class="mf">15.0</span>

  765. <span class="n">stdev_service_time</span> <span class="o">=</span> <span class="mf">3.5</span>

  766. <span class="n">num_servers</span> <span class="o">=</span> <span class="mi">3</span>

  767. 

  768. <span class="n">waits</span> <span class="o">=</span> <span class="p">[]</span>

  769. <span class="n">arrival_time</span> <span class="o">=</span> <span class="mf">0.0</span>

  770. <span class="n">servers</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.0</span><span class="p">]</span> <span class="o">*</span> <span class="n">num_servers</span>  <span class="c1"># time when each server becomes available</span>

  771. <span class="n">heapify</span><span class="p">(</span><span class="n">servers</span><span class="p">)</span>

  772. <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1_000_000</span><span class="p">):</span>

  773.     <span class="n">arrival_time</span> <span class="o">+=</span> <span class="n">expovariate</span><span class="p">(</span><span class="mf">1.0</span> <span class="o">/</span> <span class="n">average_arrival_interval</span><span class="p">)</span>

  774.     <span class="n">next_server_available</span> <span class="o">=</span> <span class="n">servers</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>

  775.     <span class="n">wait</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">next_server_available</span> <span class="o">-</span> <span class="n">arrival_time</span><span class="p">)</span>

  776.     <span class="n">waits</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">wait</span><span class="p">)</span>

  777.     <span class="n">service_duration</span> <span class="o">=</span> <span class="nb">max</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">gauss</span><span class="p">(</span><span class="n">average_service_time</span><span class="p">,</span> <span class="n">stdev_service_time</span><span class="p">))</span>

  778.     <span class="n">service_completed</span> <span class="o">=</span> <span class="n">arrival_time</span> <span class="o">+</span> <span class="n">wait</span> <span class="o">+</span> <span class="n">service_duration</span>

  779.     <span class="n">heapreplace</span><span class="p">(</span><span class="n">servers</span><span class="p">,</span> <span class="n">service_completed</span><span class="p">)</span>

  780. 

  781. <span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;Mean wait: </span><span class="si">{</span><span class="n">mean</span><span class="p">(</span><span class="n">waits</span><span class="p">)</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">   Max wait: </span><span class="si">{</span><span class="nb">max</span><span class="p">(</span><span class="n">waits</span><span class="p">)</span><span class="si">:</span><span class="s1">.1f</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">)</span>

  782. <span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Quartiles:&#39;</span><span class="p">,</span> <span class="p">[</span><span class="nb">round</span><span class="p">(</span><span class="n">q</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">quantiles</span><span class="p">(</span><span class="n">waits</span><span class="p">)])</span>

  783. </pre></div>

  784. </div>

  785. <div class="admonition seealso">

  786. <p class="admonition-title">See also</p>

  787. <p><a class="reference external" href="https://www.youtube.com/watch?v=Iq9DzN6mvYA">Statistics for Hackers</a>

  788. a video tutorial by

  789. <a class="reference external" href="https://us.pycon.org/2016/speaker/profile/295/">Jake Vanderplas</a>

  790. on statistical analysis using just a few fundamental concepts

  791. including simulation, sampling, shuffling, and cross-validation.</p>

  792. <p><a class="reference external" href="https://nbviewer.org/url/norvig.com/ipython/Economics.ipynb">Economics Simulation</a>

  793. a simulation of a marketplace by

  794. <a class="reference external" href="https://norvig.com/bio.html">Peter Norvig</a> that shows effective

  795. use of many of the tools and distributions provided by this module

  796. (gauss, uniform, sample, betavariate, choice, triangular, and randrange).</p>

  797. <p><a class="reference external" href="https://nbviewer.org/url/norvig.com/ipython/Probability.ipynb">A Concrete Introduction to Probability (using Python)</a>

  798. a tutorial by <a class="reference external" href="https://norvig.com/bio.html">Peter Norvig</a> covering

  799. the basics of probability theory, how to write simulations, and

  800. how to perform data analysis using Python.</p>

  801. </div>

  802. </section>

  803. <section id="recipes">

  804. <h2>Recipes<a class="headerlink" href="#recipes" title="Link to this heading">¶</a></h2>

  805. <p>These recipes show how to efficiently make random selections

  806. from the combinatoric iterators in the <a class="reference internal" href="itertools.html#module-itertools" title="itertools: Functions creating iterators for efficient looping."><code class="xref py py-mod docutils literal notranslate"><span class="pre">itertools</span></code></a> module:</p>

  807. <div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">random_product</span><span class="p">(</span><span class="o">*</span><span class="n">args</span><span class="p">,</span> <span class="n">repeat</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>

  808.     <span class="s2">&quot;Random selection from itertools.product(*args, **kwds)&quot;</span>

  809.     <span class="n">pools</span> <span class="o">=</span> <span class="p">[</span><span class="nb">tuple</span><span class="p">(</span><span class="n">pool</span><span class="p">)</span> <span class="k">for</span> <span class="n">pool</span> <span class="ow">in</span> <span class="n">args</span><span class="p">]</span> <span class="o">*</span> <span class="n">repeat</span>

  810.     <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="nb">map</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">,</span> <span class="n">pools</span><span class="p">))</span>

  811. 

  812. <span class="k">def</span> <span class="nf">random_permutation</span><span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">r</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>

  813.     <span class="s2">&quot;Random selection from itertools.permutations(iterable, r)&quot;</span>

  814.     <span class="n">pool</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">iterable</span><span class="p">)</span>

  815.     <span class="n">r</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pool</span><span class="p">)</span> <span class="k">if</span> <span class="n">r</span> <span class="ow">is</span> <span class="kc">None</span> <span class="k">else</span> <span class="n">r</span>

  816.     <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="n">pool</span><span class="p">,</span> <span class="n">r</span><span class="p">))</span>

  817. 

  818. <span class="k">def</span> <span class="nf">random_combination</span><span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>

  819.     <span class="s2">&quot;Random selection from itertools.combinations(iterable, r)&quot;</span>

  820.     <span class="n">pool</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">iterable</span><span class="p">)</span>

  821.     <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pool</span><span class="p">)</span>

  822.     <span class="n">indices</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">sample</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">r</span><span class="p">))</span>

  823.     <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">pool</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">)</span>

  824. 

  825. <span class="k">def</span> <span class="nf">random_combination_with_replacement</span><span class="p">(</span><span class="n">iterable</span><span class="p">,</span> <span class="n">r</span><span class="p">):</span>

  826.     <span class="s2">&quot;Choose r elements with replacement.  Order the result to match the iterable.&quot;</span>

  827.     <span class="c1"># Result will be in set(itertools.combinations_with_replacement(iterable, r)).</span>

  828.     <span class="n">pool</span> <span class="o">=</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">iterable</span><span class="p">)</span>

  829.     <span class="n">n</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">pool</span><span class="p">)</span>

  830.     <span class="n">indices</span> <span class="o">=</span> <span class="nb">sorted</span><span class="p">(</span><span class="n">random</span><span class="o">.</span><span class="n">choices</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">),</span> <span class="n">k</span><span class="o">=</span><span class="n">r</span><span class="p">))</span>

  831.     <span class="k">return</span> <span class="nb">tuple</span><span class="p">(</span><span class="n">pool</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">indices</span><span class="p">)</span>

  832. </pre></div>

  833. </div>

  834. <p>The default <a class="reference internal" href="#random.random" title="random.random"><code class="xref py py-func docutils literal notranslate"><span class="pre">random()</span></code></a> returns multiples of 2⁻⁵³ in the range

  835. <em>0.0 ≤ x &lt; 1.0</em>.  All such numbers are evenly spaced and are exactly

  836. representable as Python floats.  However, many other representable

  837. floats in that interval are not possible selections.  For example,

  838. <code class="docutils literal notranslate"><span class="pre">0.05954861408025609</span></code> isn’t an integer multiple of 2⁻⁵³.</p>

  839. <p>The following recipe takes a different approach.  All floats in the

  840. interval are possible selections.  The mantissa comes from a uniform

  841. distribution of integers in the range <em>2⁵² ≤ mantissa &lt; 2⁵³</em>.  The

  842. exponent comes from a geometric distribution where exponents smaller

  843. than <em>-53</em> occur half as often as the next larger exponent.</p>

  844. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">random</span> <span class="kn">import</span> <span class="n">Random</span>

  845. <span class="kn">from</span> <span class="nn">math</span> <span class="kn">import</span> <span class="n">ldexp</span>

  846. 

  847. <span class="k">class</span> <span class="nc">FullRandom</span><span class="p">(</span><span class="n">Random</span><span class="p">):</span>

  848. 

  849.     <span class="k">def</span> <span class="nf">random</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>

  850.         <span class="n">mantissa</span> <span class="o">=</span> <span class="mh">0x10_0000_0000_0000</span> <span class="o">|</span> <span class="bp">self</span><span class="o">.</span><span class="n">getrandbits</span><span class="p">(</span><span class="mi">52</span><span class="p">)</span>

  851.         <span class="n">exponent</span> <span class="o">=</span> <span class="o">-</span><span class="mi">53</span>

  852.         <span class="n">x</span> <span class="o">=</span> <span class="mi">0</span>

  853.         <span class="k">while</span> <span class="ow">not</span> <span class="n">x</span><span class="p">:</span>

  854.             <span class="n">x</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">getrandbits</span><span class="p">(</span><span class="mi">32</span><span class="p">)</span>

  855.             <span class="n">exponent</span> <span class="o">+=</span> <span class="n">x</span><span class="o">.</span><span class="n">bit_length</span><span class="p">()</span> <span class="o">-</span> <span class="mi">32</span>

  856.         <span class="k">return</span> <span class="n">ldexp</span><span class="p">(</span><span class="n">mantissa</span><span class="p">,</span> <span class="n">exponent</span><span class="p">)</span>

  857. </pre></div>

  858. </div>

  859. <p>All <a class="reference internal" href="#real-valued-distributions"><span class="std std-ref">real valued distributions</span></a>

  860. in the class will use the new method:</p>

  861. <div class="highlight-python3 notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">fr</span> <span class="o">=</span> <span class="n">FullRandom</span><span class="p">()</span>

  862. <span class="gp">&gt;&gt;&gt; </span><span class="n">fr</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>

  863. <span class="go">0.05954861408025609</span>

  864. <span class="gp">&gt;&gt;&gt; </span><span class="n">fr</span><span class="o">.</span><span class="n">expovariate</span><span class="p">(</span><span class="mf">0.25</span><span class="p">)</span>

  865. <span class="go">8.87925541791544</span>

  866. </pre></div>

  867. </div>

  868. <p>The recipe is conceptually equivalent to an algorithm that chooses from

  869. all the multiples of 2⁻¹⁰⁷⁴ in the range <em>0.0 ≤ x &lt; 1.0</em>.  All such

  870. numbers are evenly spaced, but most have to be rounded down to the

  871. nearest representable Python float.  (The value 2⁻¹⁰⁷⁴ is the smallest

  872. positive unnormalized float and is equal to <code class="docutils literal notranslate"><span class="pre">math.ulp(0.0)</span></code>.)</p>

  873. <div class="admonition seealso">

  874. <p class="admonition-title">See also</p>

  875. <p><a class="reference external" href="https://allendowney.com/research/rand/downey07randfloat.pdf">Generating Pseudo-random Floating-Point Values</a> a

  876. paper by Allen B. Downey describing ways to generate more

  877. fine-grained floats than normally generated by <a class="reference internal" href="#random.random" title="random.random"><code class="xref py py-func docutils literal notranslate"><span class="pre">random()</span></code></a>.</p>

  878. </div>

  879. </section>

  880. </section>

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  891.     <ul>

  892. <li><a class="reference internal" href="#"><code class="xref py py-mod docutils literal notranslate"><span class="pre">random</span></code> — Generate pseudo-random numbers</a><ul>

  893. <li><a class="reference internal" href="#bookkeeping-functions">Bookkeeping functions</a></li>

  894. <li><a class="reference internal" href="#functions-for-bytes">Functions for bytes</a></li>

  895. <li><a class="reference internal" href="#functions-for-integers">Functions for integers</a></li>

  896. <li><a class="reference internal" href="#functions-for-sequences">Functions for sequences</a></li>

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  899. <li><a class="reference internal" href="#alternative-generator">Alternative Generator</a></li>

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