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<section id="numpy-random-generator-hypergeometric">
<h1>numpy.random.Generator.hypergeometric<a class="headerlink" href="#numpy-random-generator-hypergeometric" title="Link to this heading">#</a></h1>
<p>method</p>
<dl class="py method">
<dt class="sig sig-object py" id="numpy.random.Generator.hypergeometric">
<span class="sig-prename descclassname"><span class="pre">random.Generator.</span></span><span class="sig-name descname"><span class="pre">hypergeometric</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">ngood</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nbad</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">nsample</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">size</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="#numpy.random.Generator.hypergeometric" title="Link to this definition">#</a></dt>
<dd><p>Draw samples from a Hypergeometric distribution.</p>
<p>Samples are drawn from a hypergeometric distribution with specified
parameters, <em class="xref py py-obj">ngood</em> (ways to make a good selection), <em class="xref py py-obj">nbad</em> (ways to make
a bad selection), and <em class="xref py py-obj">nsample</em> (number of items sampled, which is less
than or equal to the sum <code class="docutils literal notranslate"><span class="pre">ngood</span> <span class="pre">+</span> <span class="pre">nbad</span></code>).</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>ngood</strong><span class="classifier">int or array_like of ints</span></dt><dd><p>Number of ways to make a good selection. Must be nonnegative and
less than 10**9.</p>
</dd>
<dt><strong>nbad</strong><span class="classifier">int or array_like of ints</span></dt><dd><p>Number of ways to make a bad selection. Must be nonnegative and
less than 10**9.</p>
</dd>
<dt><strong>nsample</strong><span class="classifier">int or array_like of ints</span></dt><dd><p>Number of items sampled. Must be nonnegative and less than
<code class="docutils literal notranslate"><span class="pre">ngood</span> <span class="pre">+</span> <span class="pre">nbad</span></code>.</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape. If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n,</span> <span class="pre">k)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn. If size is <code class="docutils literal notranslate"><span class="pre">None</span></code> (default),
a single value is returned if <em class="xref py py-obj">ngood</em>, <em class="xref py py-obj">nbad</em>, and <em class="xref py py-obj">nsample</em>
are all scalars. Otherwise, <code class="docutils literal notranslate"><span class="pre">np.broadcast(ngood,</span> <span class="pre">nbad,</span> <span class="pre">nsample).size</span></code>
samples are drawn.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray or scalar</span></dt><dd><p>Drawn samples from the parameterized hypergeometric distribution. Each
sample is the number of good items within a randomly selected subset of
size <em class="xref py py-obj">nsample</em> taken from a set of <em class="xref py py-obj">ngood</em> good items and <em class="xref py py-obj">nbad</em> bad items.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.random.Generator.multivariate_hypergeometric.html#numpy.random.Generator.multivariate_hypergeometric" title="numpy.random.Generator.multivariate_hypergeometric"><code class="xref py py-obj docutils literal notranslate"><span class="pre">multivariate_hypergeometric</span></code></a></dt><dd><p>Draw samples from the multivariate hypergeometric distribution.</p>
</dd>
<dt><a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.hypergeom.html#scipy.stats.hypergeom" title="(in SciPy v1.15.2)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">scipy.stats.hypergeom</span></code></a></dt><dd><p>probability density function, distribution or cumulative density function, etc.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The probability mass function (PMF) for the Hypergeometric distribution is</p>
<div class="math notranslate nohighlight">
\[P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},\]</div>
<p>where <span class="math notranslate nohighlight">\(0 \le x \le n\)</span> and <span class="math notranslate nohighlight">\(n-b \le x \le g\)</span></p>
<p>for P(x) the probability of <code class="docutils literal notranslate"><span class="pre">x</span></code> good results in the drawn sample,
g = <em class="xref py py-obj">ngood</em>, b = <em class="xref py py-obj">nbad</em>, and n = <em class="xref py py-obj">nsample</em>.</p>
<p>Consider an urn with black and white marbles in it, <em class="xref py py-obj">ngood</em> of them
are black and <em class="xref py py-obj">nbad</em> are white. If you draw <em class="xref py py-obj">nsample</em> balls without
replacement, then the hypergeometric distribution describes the
distribution of black balls in the drawn sample.</p>
<p>Note that this distribution is very similar to the binomial
distribution, except that in this case, samples are drawn without
replacement, whereas in the Binomial case samples are drawn with
replacement (or the sample space is infinite). As the sample space
becomes large, this distribution approaches the binomial.</p>
<p>The arguments <em class="xref py py-obj">ngood</em> and <em class="xref py py-obj">nbad</em> each must be less than <em class="xref py py-obj">10**9</em>. For
extremely large arguments, the algorithm that is used to compute the
samples <a class="reference internal" href="#r688e4aa3bfc3-4" id="id1">[4]</a> breaks down because of loss of precision in floating point
calculations. For such large values, if <em class="xref py py-obj">nsample</em> is not also large,
the distribution can be approximated with the binomial distribution,
<em class="xref py py-obj">binomial(n=nsample, p=ngood/(ngood + nbad))</em>.</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="r688e4aa3bfc3-1" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>Lentner, Marvin, “Elementary Applied Statistics”, Bogden
and Quigley, 1972.</p>
</div>
<div class="citation" id="r688e4aa3bfc3-2" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<p>Weisstein, Eric W. “Hypergeometric Distribution.” From
MathWorld–A Wolfram Web Resource.
<a class="reference external" href="https://mathworld.wolfram.com/HypergeometricDistribution.html">https://mathworld.wolfram.com/HypergeometricDistribution.html</a></p>
</div>
<div class="citation" id="r688e4aa3bfc3-3" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></span>
<p>Wikipedia, “Hypergeometric distribution”,
<a class="reference external" href="https://en.wikipedia.org/wiki/Hypergeometric_distribution">https://en.wikipedia.org/wiki/Hypergeometric_distribution</a></p>
</div>
<div class="citation" id="r688e4aa3bfc3-4" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">4</a><span class="fn-bracket">]</span></span>
<p>Stadlober, Ernst, “The ratio of uniforms approach for generating
discrete random variates”, Journal of Computational and Applied
Mathematics, 31, pp. 181-189 (1990).</p>
</div>
</div>
<p class="rubric">Examples</p>
<div class="try_examples_outer_container docutils container" id="092e96f0-faed-4bec-a7a5-6a6955900d06">
<div class="try_examples_button_container"><button class="try_examples_button" onclick="window.tryExamplesShowIframe('092e96f0-faed-4bec-a7a5-6a6955900d06','c8bac9af-6aab-4551-a6c0-4f8faae0b7a0','64667c18-bc6e-462d-9043-182aebb9d27d','../../../lite/tree/../notebooks/index.html?path=5269e97b_e719_4a2b_a8a9_3966bb962302.ipynb','None')">Try it in your browser!</button></div><div class="try_examples_content docutils container">
<p>Draw samples from the distribution:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">ngood</span><span class="p">,</span> <span class="n">nbad</span><span class="p">,</span> <span class="n">nsamp</span> <span class="o">=</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">10</span>
<span class="go"># number of good, number of bad, and number of samples</span>
<span class="gp">>>> </span><span class="n">s</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">hypergeometric</span><span class="p">(</span><span class="n">ngood</span><span class="p">,</span> <span class="n">nbad</span><span class="p">,</span> <span class="n">nsamp</span><span class="p">,</span> <span class="mi">1000</span><span class="p">)</span>
<span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="kn">import</span> <span class="n">hist</span>
<span class="gp">>>> </span><span class="n">hist</span><span class="p">(</span><span class="n">s</span><span class="p">)</span>
<span class="go"># note that it is very unlikely to grab both bad items</span>
</pre></div>
</div>
<p>Suppose you have an urn with 15 white and 15 black marbles.
If you pull 15 marbles at random, how likely is it that
12 or more of them are one color?</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">hypergeometric</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">100000</span><span class="p">)</span>
<span class="gp">>>> </span><span class="nb">sum</span><span class="p">(</span><span class="n">s</span><span class="o">>=</span><span class="mi">12</span><span class="p">)</span><span class="o">/</span><span class="mf">100000.</span> <span class="o">+</span> <span class="nb">sum</span><span class="p">(</span><span class="n">s</span><span class="o"><=</span><span class="mi">3</span><span class="p">)</span><span class="o">/</span><span class="mf">100000.</span>
<span class="go"># answer = 0.003 ... pretty unlikely!</span>
</pre></div>
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