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 <div class="section" id="module-random">
<h1>9.6. <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">random</code></a> — Generate pseudo-random numbers<a class="headerlink" href="#module-random" title="Permalink to this headline">¶</a></h1>
Source code: <a class="reference external" href="https://github.com/python/cpython/tree/3.6/Lib/random.py">Lib/random.py</a>
<hr class="docutils" />
This module implements pseudo-random number generators for various
distributions.
For integers, there is uniform selection from a range. For sequences, there is
uniform selection of a random element, a function to generate a random
permutation of a list in-place, and a function for random sampling without
replacement.
On the real line, there are functions to compute uniform, normal (Gaussian),
lognormal, negative exponential, gamma, and beta distributions. For generating
distributions of angles, the von Mises distribution is available.
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">random()</code></a>, which
generates a random float uniformly in the semi-open range [0.0, 1.0). Python
uses the Mersenne Twister as the core generator. It produces 53-bit precision
floats and has a period of 2**19937-1. The underlying implementation in C is
both fast and threadsafe. The Mersenne Twister is one of the most extensively
tested random number generators in existence. However, being completely
deterministic, it is not suitable for all purposes, and is completely unsuitable
for cryptographic purposes.
The functions supplied by this module are actually bound methods of a hidden
instance of the <code class="xref py py-class docutils literal notranslate">random.Random</code> class. You can instantiate your own
instances of <code class="xref py py-class docutils literal notranslate">Random</code> to get generators that don’t share state.
Class <code class="xref py py-class docutils literal notranslate">Random</code> can also be subclassed if you want to use a different
basic generator of your own devising: in that case, override the <code class="xref py py-meth docutils literal notranslate">random()</code>,
<code class="xref py py-meth docutils literal notranslate">seed()</code>, <code class="xref py py-meth docutils literal notranslate">getstate()</code>, and <code class="xref py py-meth docutils literal notranslate">setstate()</code> methods.
Optionally, a new generator can supply a <code class="xref py py-meth docutils literal notranslate">getrandbits()</code> method — this
allows <a class="reference internal" href="#random.randrange" title="random.randrange"><code class="xref py py-meth docutils literal notranslate">randrange()</code></a> to produce selections over an arbitrarily large range.
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">random</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">SystemRandom</code></a> class which
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">os.urandom()</code></a> to generate random numbers
from sources provided by the operating system.
<div class="admonition warning">
Warning
The pseudo-random generators of this module should not be used for
security purposes. For security or cryptographic uses, see the
<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">secrets</code></a> module.
</div>
<div class="admonition seealso">
See also
M. Matsumoto and T. Nishimura, “Mersenne Twister: A 623-dimensionally
equidistributed uniform pseudorandom number generator”, ACM Transactions on
Modeling and Computer Simulation Vol. 8, No. 1, January pp.3–30 1998.
<a class="reference external" href="https://code.activestate.com/recipes/576707/">Complementary-Multiply-with-Carry recipe</a> for a compatible alternative
random number generator with a long period and comparatively simple update
operations.
</div>
<div class="section" id="bookkeeping-functions">
<h2>9.6.1. Bookkeeping functions<a class="headerlink" href="#bookkeeping-functions" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="random.seed">
<code class="descclassname">random.</code><code class="descname">seed</code>(a=None, version=2)<a class="headerlink" href="#random.seed" title="Permalink to this definition">¶</a></dt>
<dd>Initialize the random number generator.
If a is omitted or <code class="docutils literal notranslate">None</code>, the current system time is used. If
randomness sources are provided by the operating system, they are used
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">os.urandom()</code></a> function for details
on availability).
If a is an int, it is used directly.
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">str</code></a>, <a class="reference internal" href="stdtypes.html#bytes" title="bytes"><code class="xref py py-class docutils literal notranslate">bytes</code></a>, or <a class="reference internal" href="stdtypes.html#bytearray" title="bytearray"><code class="xref py py-class docutils literal notranslate">bytearray</code></a>
object gets converted to an <a class="reference internal" href="functions.html#int" title="int"><code class="xref py py-class docutils literal notranslate">int</code></a> and all of its bits are used.
With version 1 (provided for reproducing random sequences from older versions
of Python), the algorithm for <a class="reference internal" href="stdtypes.html#str" title="str"><code class="xref py py-class docutils literal notranslate">str</code></a> and <a class="reference internal" href="stdtypes.html#bytes" title="bytes"><code class="xref py py-class docutils literal notranslate">bytes</code></a> generates a
narrower range of seeds.
<div class="versionchanged">
Changed in version 3.2: Moved to the version 2 scheme which uses all of the bits in a string seed.
</div>
</dd></dl>

<dl class="function">
<dt id="random.getstate">
<code class="descclassname">random.</code><code class="descname">getstate</code>()<a class="headerlink" href="#random.getstate" title="Permalink to this definition">¶</a></dt>
<dd>Return an object capturing the current internal state of the generator. This
object can be passed to <a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-func docutils literal notranslate">setstate()</code></a> to restore the state.
</dd></dl>

<dl class="function">
<dt id="random.setstate">
<code class="descclassname">random.</code><code class="descname">setstate</code>(state)<a class="headerlink" href="#random.setstate" title="Permalink to this definition">¶</a></dt>
<dd>state 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">getstate()</code></a>, and
<a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-func docutils literal notranslate">setstate()</code></a> restores the internal state of the generator to what it was at
the time <a class="reference internal" href="#random.getstate" title="random.getstate"><code class="xref py py-func docutils literal notranslate">getstate()</code></a> was called.
</dd></dl>

<dl class="function">
<dt id="random.getrandbits">
<code class="descclassname">random.</code><code class="descname">getrandbits</code>(k)<a class="headerlink" href="#random.getrandbits" title="Permalink to this definition">¶</a></dt>
<dd>Returns a Python integer with k random bits. This method is supplied with
the MersenneTwister generator and some other generators may also provide it
as an optional part of the API. When available, <a class="reference internal" href="#random.getrandbits" title="random.getrandbits"><code class="xref py py-meth docutils literal notranslate">getrandbits()</code></a> enables
<a class="reference internal" href="#random.randrange" title="random.randrange"><code class="xref py py-meth docutils literal notranslate">randrange()</code></a> to handle arbitrarily large ranges.
</dd></dl>

</div>
<div class="section" id="functions-for-integers">
<h2>9.6.2. Functions for integers<a class="headerlink" href="#functions-for-integers" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="random.randrange">
<code class="descclassname">random.</code><code class="descname">randrange</code>(stop)<a class="headerlink" href="#random.randrange" title="Permalink to this definition">¶</a></dt>
<dt>
<code class="descclassname">random.</code><code class="descname">randrange</code>(start, stop[, step])</dt>
<dd>Return a randomly selected element from <code class="docutils literal notranslate">range(start, stop, step)</code>. This is
equivalent to <code class="docutils literal notranslate">choice(range(start, stop, step))</code>, but doesn’t actually build a
range object.
The positional argument pattern matches that of <a class="reference internal" href="stdtypes.html#range" title="range"><code class="xref py py-func docutils literal notranslate">range()</code></a>. Keyword arguments
should not be used because the function may use them in unexpected ways.
<div class="versionchanged">
Changed in version 3.2: <a class="reference internal" href="#random.randrange" title="random.randrange"><code class="xref py py-meth docutils literal notranslate">randrange()</code></a> is more sophisticated about producing equally distributed
values. Formerly it used a style like <code class="docutils literal notranslate">int(random()*n)</code> which could produce
slightly uneven distributions.
</div>
</dd></dl>

<dl class="function">
<dt id="random.randint">
<code class="descclassname">random.</code><code class="descname">randint</code>(a, b)<a class="headerlink" href="#random.randint" title="Permalink to this definition">¶</a></dt>
<dd>Return a random integer N such that <code class="docutils literal notranslate">a &lt;= N &lt;= b</code>. Alias for
<code class="docutils literal notranslate">randrange(a, b+1)</code>.
</dd></dl>

</div>
<div class="section" id="functions-for-sequences">
<h2>9.6.3. Functions for sequences<a class="headerlink" href="#functions-for-sequences" title="Permalink to this headline">¶</a></h2>
<dl class="function">
<dt id="random.choice">
<code class="descclassname">random.</code><code class="descname">choice</code>(seq)<a class="headerlink" href="#random.choice" title="Permalink to this definition">¶</a></dt>
<dd>Return a random element from the non-empty sequence seq. If seq is empty,
raises <a class="reference internal" href="exceptions.html#IndexError" title="IndexError"><code class="xref py py-exc docutils literal notranslate">IndexError</code></a>.
</dd></dl>

<dl class="function">
<dt id="random.choices">
<code class="descclassname">random.</code><code class="descname">choices</code>(population, weights=None, *, cum_weights=None, k=1)<a class="headerlink" href="#random.choices" title="Permalink to this definition">¶</a></dt>
<dd>Return a k sized list of elements chosen from the population with replacement.
If the population is empty, raises <a class="reference internal" href="exceptions.html#IndexError" title="IndexError"><code class="xref py py-exc docutils literal notranslate">IndexError</code></a>.
If a weights sequence is specified, selections are made according to the
relative weights. Alternatively, if a cum_weights sequence is given, the
selections are made according to the cumulative weights (perhaps computed
using <a class="reference internal" href="itertools.html#itertools.accumulate" title="itertools.accumulate"><code class="xref py py-func docutils literal notranslate">itertools.accumulate()</code></a>). For example, the relative weights
<code class="docutils literal notranslate">[10, 5, 30, 5]</code> are equivalent to the cumulative weights
<code class="docutils literal notranslate">[10, 15, 45, 50]</code>. Internally, the relative weights are converted to
cumulative weights before making selections, so supplying the cumulative
weights saves work.
If neither weights nor cum_weights are specified, selections are made
with equal probability. If a weights sequence is supplied, it must be
the same length as the population sequence. It is a <a class="reference internal" href="exceptions.html#TypeError" title="TypeError"><code class="xref py py-exc docutils literal notranslate">TypeError</code></a>
to specify both weights and cum_weights.
The weights or cum_weights can use any numeric type that interoperates
with the <a class="reference internal" href="functions.html#float" title="float"><code class="xref py py-class docutils literal notranslate">float</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">random()</code></a> (that includes
integers, floats, and fractions but excludes decimals).
<div class="versionadded">
New in version 3.6.
</div>
</dd></dl>

<dl class="function">
<dt id="random.shuffle">
<code class="descclassname">random.</code><code class="descname">shuffle</code>(x[, random])<a class="headerlink" href="#random.shuffle" title="Permalink to this definition">¶</a></dt>
<dd>Shuffle the sequence x in place.
The optional argument random is a 0-argument function returning a random
float in [0.0, 1.0); by default, this is the function <a class="reference internal" href="#random.random" title="random.random"><code class="xref py py-func docutils literal notranslate">random()</code></a>.
To shuffle an immutable sequence and return a new shuffled list, use
<code class="docutils literal notranslate">sample(x, k=len(x))</code> instead.
Note that even for small <code class="docutils literal notranslate">len(x)</code>, the total number of permutations of x
can quickly grow larger than the period of most random number generators.
This implies that most permutations of a long sequence can never be
generated. For example, a sequence of length 2080 is the largest that
can fit within the period of the Mersenne Twister random number generator.
</dd></dl>

<dl class="function">
<dt id="random.sample">
<code class="descclassname">random.</code><code class="descname">sample</code>(population, k)<a class="headerlink" href="#random.sample" title="Permalink to this definition">¶</a></dt>
<dd>Return a k length list of unique elements chosen from the population sequence
or set. Used for random sampling without replacement.
Returns a new list containing elements from the population while leaving the
original population unchanged. The resulting list is in selection order so that
all sub-slices will also be valid random samples. This allows raffle winners
(the sample) to be partitioned into grand prize and second place winners (the
subslices).
Members of the population need not be <a class="reference internal" href="../glossary.html#term-hashable">hashable</a> or unique. If the population
contains repeats, then each occurrence is a possible selection in the sample.
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">range()</code></a> object as an
argument. This is especially fast and space efficient for sampling from a large
population: <code class="docutils literal notranslate">sample(range(10000000), k=60)</code>.
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">ValueError</code></a>
is raised.
</dd></dl>

</div>
<div class="section" id="real-valued-distributions">
<h2>9.6.4. Real-valued distributions<a class="headerlink" href="#real-valued-distributions" title="Permalink to this headline">¶</a></h2>
The following functions generate specific real-valued distributions. Function
parameters are named after the corresponding variables in the distribution’s
equation, as used in common mathematical practice; most of these equations can
be found in any statistics text.
<dl class="function">
<dt id="random.random">
<code class="descclassname">random.</code><code class="descname">random</code>()<a class="headerlink" href="#random.random" title="Permalink to this definition">¶</a></dt>
<dd>Return the next random floating point number in the range [0.0, 1.0).
</dd></dl>

<dl class="function">
<dt id="random.uniform">
<code class="descclassname">random.</code><code class="descname">uniform</code>(a, b)<a class="headerlink" href="#random.uniform" title="Permalink to this definition">¶</a></dt>
<dd>Return a random floating point number N such that <code class="docutils literal notranslate">a &lt;= N &lt;= b</code> for
<code class="docutils literal notranslate">a &lt;= b</code> and <code class="docutils literal notranslate">b &lt;= N &lt;= a</code> for <code class="docutils literal notranslate">b &lt; a</code>.
The end-point value <code class="docutils literal notranslate">b</code> may or may not be included in the range
depending on floating-point rounding in the equation <code class="docutils literal notranslate">a + (b-a) * random()</code>.
</dd></dl>

<dl class="function">
<dt id="random.triangular">
<code class="descclassname">random.</code><code class="descname">triangular</code>(low, high, mode)<a class="headerlink" href="#random.triangular" title="Permalink to this definition">¶</a></dt>
<dd>Return a random floating point number N such that <code class="docutils literal notranslate">low &lt;= N &lt;= high</code> and
with the specified mode between those bounds. The low and high bounds
default to zero and one. The mode argument defaults to the midpoint
between the bounds, giving a symmetric distribution.
</dd></dl>

<dl class="function">
<dt id="random.betavariate">
<code class="descclassname">random.</code><code class="descname">betavariate</code>(alpha, beta)<a class="headerlink" href="#random.betavariate" title="Permalink to this definition">¶</a></dt>
<dd>Beta distribution. Conditions on the parameters are <code class="docutils literal notranslate">alpha &gt; 0</code> and
<code class="docutils literal notranslate">beta &gt; 0</code>. Returned values range between 0 and 1.
</dd></dl>

<dl class="function">
<dt id="random.expovariate">
<code class="descclassname">random.</code><code class="descname">expovariate</code>(lambd)<a class="headerlink" href="#random.expovariate" title="Permalink to this definition">¶</a></dt>
<dd>Exponential distribution. lambd is 1.0 divided by the desired
mean. It should be nonzero. (The parameter would be called
“lambda”, but that is a reserved word in Python.) Returned values
range from 0 to positive infinity if lambd is positive, and from
negative infinity to 0 if lambd is negative.
</dd></dl>

<dl class="function">
<dt id="random.gammavariate">
<code class="descclassname">random.</code><code class="descname">gammavariate</code>(alpha, beta)<a class="headerlink" href="#random.gammavariate" title="Permalink to this definition">¶</a></dt>
<dd>Gamma distribution. (Not the gamma function!) Conditions on the
parameters are <code class="docutils literal notranslate">alpha &gt; 0</code> and <code class="docutils literal notranslate">beta &gt; 0</code>.
The probability distribution function is:
<div class="highlight-python3 notranslate"><div class="highlight"><pre> x ** (alpha - 1) * math.exp(-x / beta)
pdf(x) = --------------------------------------
 math.gamma(alpha) * beta ** alpha
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="random.gauss">
<code class="descclassname">random.</code><code class="descname">gauss</code>(mu, sigma)<a class="headerlink" href="#random.gauss" title="Permalink to this definition">¶</a></dt>
<dd>Gaussian distribution. mu is the mean, and sigma is the standard
deviation. This is slightly faster than the <a class="reference internal" href="#random.normalvariate" title="random.normalvariate"><code class="xref py py-func docutils literal notranslate">normalvariate()</code></a> function
defined below.
</dd></dl>

<dl class="function">
<dt id="random.lognormvariate">
<code class="descclassname">random.</code><code class="descname">lognormvariate</code>(mu, sigma)<a class="headerlink" href="#random.lognormvariate" title="Permalink to this definition">¶</a></dt>
<dd>Log normal distribution. If you take the natural logarithm of this
distribution, you’ll get a normal distribution with mean mu and standard
deviation sigma. mu can have any value, and sigma must be greater than
zero.
</dd></dl>

<dl class="function">
<dt id="random.normalvariate">
<code class="descclassname">random.</code><code class="descname">normalvariate</code>(mu, sigma)<a class="headerlink" href="#random.normalvariate" title="Permalink to this definition">¶</a></dt>
<dd>Normal distribution. mu is the mean, and sigma is the standard deviation.
</dd></dl>

<dl class="function">
<dt id="random.vonmisesvariate">
<code class="descclassname">random.</code><code class="descname">vonmisesvariate</code>(mu, kappa)<a class="headerlink" href="#random.vonmisesvariate" title="Permalink to this definition">¶</a></dt>
<dd>mu is the mean angle, expressed in radians between 0 and 2*pi, and kappa
is the concentration parameter, which must be greater than or equal to zero. If
kappa is equal to zero, this distribution reduces to a uniform random angle
over the range 0 to 2*pi.
</dd></dl>

<dl class="function">
<dt id="random.paretovariate">
<code class="descclassname">random.</code><code class="descname">paretovariate</code>(alpha)<a class="headerlink" href="#random.paretovariate" title="Permalink to this definition">¶</a></dt>
<dd>Pareto distribution. alpha is the shape parameter.
</dd></dl>

<dl class="function">
<dt id="random.weibullvariate">
<code class="descclassname">random.</code><code class="descname">weibullvariate</code>(alpha, beta)<a class="headerlink" href="#random.weibullvariate" title="Permalink to this definition">¶</a></dt>
<dd>Weibull distribution. alpha is the scale parameter and beta is the shape
parameter.
</dd></dl>

</div>
<div class="section" id="alternative-generator">
<h2>9.6.5. Alternative Generator<a class="headerlink" href="#alternative-generator" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="random.SystemRandom">
class <code class="descclassname">random.</code><code class="descname">SystemRandom</code>([seed])<a class="headerlink" href="#random.SystemRandom" title="Permalink to this definition">¶</a></dt>
<dd>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">os.urandom()</code></a> function for generating random numbers
from sources provided by the operating system. Not available on all systems.
Does not rely on software state, and sequences are not reproducible. Accordingly,
the <a class="reference internal" href="#random.seed" title="random.seed"><code class="xref py py-meth docutils literal notranslate">seed()</code></a> method has no effect and is ignored.
The <a class="reference internal" href="#random.getstate" title="random.getstate"><code class="xref py py-meth docutils literal notranslate">getstate()</code></a> and <a class="reference internal" href="#random.setstate" title="random.setstate"><code class="xref py py-meth docutils literal notranslate">setstate()</code></a> methods raise
<a class="reference internal" href="exceptions.html#NotImplementedError" title="NotImplementedError"><code class="xref py py-exc docutils literal notranslate">NotImplementedError</code></a> if called.
</dd></dl>

</div>
<div class="section" id="notes-on-reproducibility">
<h2>9.6.6. Notes on Reproducibility<a class="headerlink" href="#notes-on-reproducibility" title="Permalink to this headline">¶</a></h2>
Sometimes it is useful to be able to reproduce the sequences given by a pseudo
random number generator. By re-using a seed value, the same sequence should be
reproducible from run to run as long as multiple threads are not running.
Most of the random module’s algorithms and seeding functions are subject to
change across Python versions, but two aspects are guaranteed not to change:
<ul class="simple">
<li>If a new seeding method is added, then a backward compatible seeder will be
offered.</li>
<li>The generator’s <code class="xref py py-meth docutils literal notranslate">random()</code> method will continue to produce the same
sequence when the compatible seeder is given the same seed.</li>
</ul>
</div>
<div class="section" id="examples-and-recipes">
<h2>9.6.7. Examples and Recipes<a class="headerlink" href="#examples-and-recipes" title="Permalink to this headline">¶</a></h2>
Basic examples:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; random() # Random float: 0.0 &lt;= x &lt; 1.0
0.37444887175646646

&gt;&gt;&gt; uniform(2.5, 10.0) # Random float: 2.5 &lt;= x &lt; 10.0
3.1800146073117523

&gt;&gt;&gt; expovariate(1 / 5) # Interval between arrivals averaging 5 seconds
5.148957571865031

&gt;&gt;&gt; randrange(10) # Integer from 0 to 9 inclusive
7

&gt;&gt;&gt; randrange(0, 101, 2) # Even integer from 0 to 100 inclusive
26

&gt;&gt;&gt; choice([&#39;win&#39;, &#39;lose&#39;, &#39;draw&#39;]) # Single random element from a sequence
&#39;draw&#39;

&gt;&gt;&gt; deck = &#39;ace two three four&#39;.split()
&gt;&gt;&gt; shuffle(deck) # Shuffle a list
&gt;&gt;&gt; deck
[&#39;four&#39;, &#39;two&#39;, &#39;ace&#39;, &#39;three&#39;]

&gt;&gt;&gt; sample([10, 20, 30, 40, 50], k=4) # Four samples without replacement
[40, 10, 50, 30]
</pre></div>
</div>
Simulations:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; # Six roulette wheel spins (weighted sampling with replacement)
&gt;&gt;&gt; choices([&#39;red&#39;, &#39;black&#39;, &#39;green&#39;], [18, 18, 2], k=6)
[&#39;red&#39;, &#39;green&#39;, &#39;black&#39;, &#39;black&#39;, &#39;red&#39;, &#39;black&#39;]

&gt;&gt;&gt; # Deal 20 cards without replacement from a deck of 52 playing cards
&gt;&gt;&gt; # and determine the proportion of cards with a ten-value
&gt;&gt;&gt; # (a ten, jack, queen, or king).
&gt;&gt;&gt; deck = collections.Counter(tens=16, low_cards=36)
&gt;&gt;&gt; seen = sample(list(deck.elements()), k=20)
&gt;&gt;&gt; seen.count(&#39;tens&#39;) / 20
0.15

&gt;&gt;&gt; # Estimate the probability of getting 5 or more heads from 7 spins
&gt;&gt;&gt; # of a biased coin that settles on heads 60% of the time.
&gt;&gt;&gt; trial = lambda: choices(&#39;HT&#39;, cum_weights=(0.60, 1.00), k=7).count(&#39;H&#39;) &gt;= 5
&gt;&gt;&gt; sum(trial() for i in range(10000)) / 10000
0.4169

&gt;&gt;&gt; # Probability of the median of 5 samples being in middle two quartiles
&gt;&gt;&gt; trial = lambda : 2500 &lt;= sorted(choices(range(10000), k=5))[2] &lt; 7500
&gt;&gt;&gt; sum(trial() for i in range(10000)) / 10000
0.7958
</pre></div>
</div>
Example of <a class="reference external" href="https://en.wikipedia.org/wiki/Bootstrapping_(statistics)">statistical bootstrapping</a> using resampling
with replacement to estimate a confidence interval for the mean of a sample of
size five:
<div class="highlight-python3 notranslate"><div class="highlight"><pre># http://statistics.about.com/od/Applications/a/Example-Of-Bootstrapping.htm
from statistics import mean
from random import choices

data = 1, 2, 4, 4, 10
means = sorted(mean(choices(data, k=5)) for i in range(20))
print(f&#39;The sample mean of {mean(data):.1f} has a 90% confidence &#39;
 f&#39;interval from {means[1]:.1f} to {means[-2]:.1f}&#39;)
</pre></div>
</div>
Example of a <a class="reference external" href="https://en.wikipedia.org/wiki/Resampling_(statistics)#Permutation_tests">resampling permutation test</a>
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
between the effects of a drug versus a placebo:
<div class="highlight-python3 notranslate"><div class="highlight"><pre># Example from &quot;Statistics is Easy&quot; by Dennis Shasha and Manda Wilson
from statistics import mean
from random import shuffle

drug = [54, 73, 53, 70, 73, 68, 52, 65, 65]
placebo = [54, 51, 58, 44, 55, 52, 42, 47, 58, 46]
observed_diff = mean(drug) - mean(placebo)

n = 10000
count = 0
combined = drug + placebo
for i in range(n):
 shuffle(combined)
 new_diff = mean(combined[:len(drug)]) - mean(combined[len(drug):])
 count += (new_diff &gt;= observed_diff)

print(f&#39;{n} label reshufflings produced only {count} instances with a difference&#39;)
print(f&#39;at least as extreme as the observed difference of {observed_diff:.1f}.&#39;)
print(f&#39;The one-sided p-value of {count / n:.4f} leads us to reject the null&#39;)
print(f&#39;hypothesis that there is no difference between the drug and the placebo.&#39;)
</pre></div>
</div>
Simulation of arrival times and service deliveries in a single server queue:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>from random import expovariate, gauss
from statistics import mean, median, stdev

average_arrival_interval = 5.6
average_service_time = 5.0
stdev_service_time = 0.5

num_waiting = 0
arrivals = []
starts = []
arrival = service_end = 0.0
for i in range(20000):
 if arrival &lt;= service_end:
 num_waiting += 1
 arrival += expovariate(1.0 / average_arrival_interval)
 arrivals.append(arrival)
 else:
 num_waiting -= 1
 service_start = service_end if num_waiting else arrival
 service_time = gauss(average_service_time, stdev_service_time)
 service_end = service_start + service_time
 starts.append(service_start)

waits = [start - arrival for arrival, start in zip(arrivals, starts)]
print(f&#39;Mean wait: {mean(waits):.1f}. Stdev wait: {stdev(waits):.1f}.&#39;)
print(f&#39;Median wait: {median(waits):.1f}. Max wait: {max(waits):.1f}.&#39;)
</pre></div>
</div>
<div class="admonition seealso">
See also
<a class="reference external" href="https://www.youtube.com/watch?v=Iq9DzN6mvYA">Statistics for Hackers</a>
a video tutorial by
<a class="reference external" href="https://us.pycon.org/2016/speaker/profile/295/">Jake Vanderplas</a>
on statistical analysis using just a few fundamental concepts
including simulation, sampling, shuffling, and cross-validation.
<a class="reference external" href="http://nbviewer.jupyter.org/url/norvig.com/ipython/Economics.ipynb">Economics Simulation</a>
a simulation of a marketplace by
<a class="reference external" href="http://norvig.com/bio.html">Peter Norvig</a> that shows effective
use of many of the tools and distributions provided by this module
(gauss, uniform, sample, betavariate, choice, triangular, and randrange).
<a class="reference external" href="http://nbviewer.jupyter.org/url/norvig.com/ipython/Probability.ipynb">A Concrete Introduction to Probability (using Python)</a>
a tutorial by <a class="reference external" href="http://norvig.com/bio.html">Peter Norvig</a> covering
the basics of probability theory, how to write simulations, and
how to perform data analysis using Python.
</div>
</div>
</div>

</div>
 </div>
 </div>
 <div class="sphinxsidebar" role="navigation" aria-label="main navigation">
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 <h3><a href="../contents.html">Table Of Contents</a></h3>
 <ul>
<li><a class="reference internal" href="#">9.6. <code class="docutils literal notranslate">random</code> — Generate pseudo-random numbers</a><ul>
<li><a class="reference internal" href="#bookkeeping-functions">9.6.1. Bookkeeping functions</a></li>
<li><a class="reference internal" href="#functions-for-integers">9.6.2. Functions for integers</a></li>
<li><a class="reference internal" href="#functions-for-sequences">9.6.3. Functions for sequences</a></li>
<li><a class="reference internal" href="#real-valued-distributions">9.6.4. Real-valued distributions</a></li>
<li><a class="reference internal" href="#alternative-generator">9.6.5. Alternative Generator</a></li>
<li><a class="reference internal" href="#notes-on-reproducibility">9.6.6. Notes on Reproducibility</a></li>
<li><a class="reference internal" href="#examples-and-recipes">9.6.7. Examples and Recipes</a></li>
</ul>
</li>
</ul>

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