# Chapter 6 Probability

## Conditional Probability

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In [1]:

import random

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In [2]:

def random_kid():
return random.choice(['girl', 'boy'])

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In [3]:

both_girls = 0
older_girl = 0
either_girl = 0

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In [4]:

random.seed(0)

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In [5]:

for _ in range(10000):
younger_kid = random_kid()
older_kid = random_kid()
if older_kid == 'girl':
older_girl += 1
if younger_kid == 'girl' and older_kid == 'girl':
both_girls += 1
if younger_kid == 'girl' or older_kid == 'girl':
either_girl += 1

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In [6]:

print('P(both | older) =', both_girls / older_girl)

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P(both | older) = 0.5008888011060636

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In [7]:

print('P(both | either) =', both_girls / either_girl)

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P(both | either) = 0.3368756641870351

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## Continuous Distribution

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In [8]:

def uniform_pdf(x):
return 1 if x >= 0 and x < 1 else 0

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In [9]:

def uniform_cdf(x):
"""returns the probability that a uniform random variable is <= x"""
if x < 0:
return 0
elif x < 1:
return x
else:
return 1

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## The Normal Distribution

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In [10]:

import math

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In [11]:

def normal_pdf(x, mu=0, sigma=1):
sqrt_two_pi = math.sqrt(2 * math.pi)
return math.exp(-(x - mu) ** 2 / (2 * sigma ** 2)) / (sqrt_two_pi * sigma)

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In [12]:

%matplotlib inline
import matplotlib.pyplot as plt

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In [13]:

xs = [x / 10.0 for x in range(-50, 50)]

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In [14]:

plt.plot(xs, [normal_pdf(x) for x in xs], '-', label='mu=0, sigma=1')
plt.plot(xs, [normal_pdf(x, sigma=2) for x in xs], '--', label='mu=0, sigma=2')
plt.plot(xs, [normal_pdf(x, sigma=0.5) for x in xs], ':', label='mu=0, sigma=0.5')
plt.plot(xs, [normal_pdf(x, mu=-1) for x in xs], '-.', label='mu=-1, sigma=1')
plt.legend()
plt.title('Various Normal pdfs')
plt.show()

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In [15]:

def normal_cdf(x, mu=0, sigma=1):
return (1 + math.erf((x - mu) / math.sqrt(2) / sigma)) / 2

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In [16]:

plt.plot(xs, [normal_cdf(x) for x in xs], '-', label='mu=0, sigma=1')
plt.plot(xs, [normal_cdf(x, sigma=2) for x in xs], '--', label='mu=0, sigma=2')
plt.plot(xs, [normal_cdf(x, sigma=0.5) for x in xs], ':', label='mu=0, sigma=0.5')
plt.plot(xs, [normal_cdf(x, mu=-1) for x in xs], '-.', label='mu=-1, sigma=1')
plt.legend(loc=4)
plt.title('Various Normal cdfs')
plt.show()

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In [17]:

def inverse_normal_cdf(p, mu=0, sigma=1, tolerance=0.00001):
"""find approximate inverse using binary search"""

# if not standard, compute standard and rescale
if mu != 0 or sigma != 1:
return mu + sigma * inverse_normal_cdf(p, tolerance=tolerance)

low_z, low_p = -10.0, 0
hi_z, hi_p = 10.0, 1
while hi_z - low_z > tolerance:
mid_z = (low_z + hi_z) / 2
mid_p = normal_cdf(mid_z)
if mid_p < p:
low_z = mid_z
low_p = mid_p
elif mid_p > p:
hi_z = mid_z
hi_p = mid_p
else:
break

return mid_z

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## The Central Limit Theorem

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In [18]:

import random
from collections import Counter

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In [19]:

def bernoulli_trial(p):
return 1 if random.random() < p else 0

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In [20]:

def binomial(n, p):
return sum(bernoulli_trial(p) for _ in range(n))

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In [21]:

def make_hist(p, n, num_points):

data = [binomial(n, p) for _ in range(num_points)]

# use a bar chart to show the actual binomial samples
histogram = Counter(data)
plt.bar([x - 0.4 for x in histogram.keys()],
[v / num_points for v in histogram.values()],
0.8,
color='0.75')

mu = p * n
sigma = math.sqrt(n * p * (1 - p))

# use a line chart to show the normal approximation
xs = range(min(data), max(data) + 1)
ys = [normal_cdf(i + 0.5, mu, sigma) - normal_cdf(i - 0.5, mu, sigma)
for i in xs]
plt.plot(xs, ys)
plt.title('Binomial Distribution vs. Normal Approximation')
plt.show()

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In [24]:

make_hist(0.75, 100, 10000)

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