From 0459ecf765bfe66d210b311af591a26a700a6495 Mon Sep 17 00:00:00 2001
From: Raymond Hettinger <raymond.hettinger@gmail.com>
Date: Sat, 25 Jan 2020 19:25:53 -0800
Subject: [PATCH 1/2] bpo-36018: Add another example for NormalDist()

---
 Doc/library/statistics.rst | 36 ++++++++++++++++++++++++++++++++++++
 1 file changed, 36 insertions(+)

diff --git a/Doc/library/statistics.rst b/Doc/library/statistics.rst
index 4c7239c1895fbf..86f166c90db017 100644
--- a/Doc/library/statistics.rst
+++ b/Doc/library/statistics.rst
@@ -772,6 +772,42 @@ Carlo simulation <https://en.wikipedia.org/wiki/Monte_Carlo_method>`_:
     >>> quantiles(map(model, X, Y, Z))       # doctest: +SKIP
     [1.4591308524824727, 1.8035946855390597, 2.175091447274739]
 
+Normal distributions can be used to approximate `Binomial
+distributions <http://mathworld.wolfram.com/BinomialDistribution.html>`
+when the sample size is large and when the probability of a successful
+trial is near 50%.
+
+For example, an open source conference has 750 attendees and two rooms with a
+500 person capacity.  There is a talk about Python and another about Ruby.
+In previous conferences, 65% of the attendees preferred to listen to Python
+talks.  Assuming the population preferences haven't changed, what is the
+probability that the rooms will stay within their capacity limits?
+
+.. doctest::
+
+    >>> n = 750             # Sample size
+    >>> p = 0.65            # Preference for Python
+    >>> q = 1.0 - p         # Preference for Ruby
+    >>> k = 500             # Room capacity
+
+    >>> # Approximation using the cumulative normal distribution
+    >>> from math import sqrt
+    >>> round(NormalDist(mu=n*p, sigma=sqrt(n*p*q)).cdf(k + 0.5), 4)
+    0.8402
+
+    >>> # Solution using the cumulative binomial distribution
+    >>> from math import comb, fsum
+    >>> round(fsum(comb(n, r) * p**r * q**(n-r) for r in range(k+1)), 4)
+    0.8402
+
+    >>> # Approximation using a simulation
+    >>> from random import seed, choices
+    >>> seed(8675309)
+    >>> def trial():
+    ...     return choices(('Python', 'Ruby'), (p, q), k=n).count('Python')
+    >>> mean(trial() <= k for i in range(10_000))
+    0.8398
+
 Normal distributions commonly arise in machine learning problems.
 
 Wikipedia has a `nice example of a Naive Bayesian Classifier

From 2c959620f44f6ba4fd5163baea6a915365dc2db2 Mon Sep 17 00:00:00 2001
From: Raymond Hettinger <raymond.hettinger@gmail.com>
Date: Sat, 25 Jan 2020 19:44:21 -0800
Subject: [PATCH 2/2] Fix markup

---
 Doc/library/statistics.rst | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Doc/library/statistics.rst b/Doc/library/statistics.rst
index 86f166c90db017..09b02cabf21f8e 100644
--- a/Doc/library/statistics.rst
+++ b/Doc/library/statistics.rst
@@ -773,7 +773,7 @@ Carlo simulation <https://en.wikipedia.org/wiki/Monte_Carlo_method>`_:
     [1.4591308524824727, 1.8035946855390597, 2.175091447274739]
 
 Normal distributions can be used to approximate `Binomial
-distributions <http://mathworld.wolfram.com/BinomialDistribution.html>`
+distributions <http://mathworld.wolfram.com/BinomialDistribution.html>`_
 when the sample size is large and when the probability of a successful
 trial is near 50%.