We can use the binom.cdf() method from the scipy.stats library to calculate the cumulative distribution function. This method takes 3 values:

• x: the value of interest, looking for the probability of this value or less
• n: the sample size
• p: the probability of success

Calculating the probability of observing 6 or fewer heads from 10 fair coin flips (0 to 6 heads) mathematically looks like the following:

$P(6\; or\; fewer\; heads) = P(0\; to\; 6\; heads)$

In python, we use the following code:

import scipy.stats as stats

print(stats.binom.cdf(6, 10, 0.5))

Output:

0.828125

Calculating the probability of observing between 4 and 8 heads from 10 fair coin flips can be thought of as taking the difference of the value of the cumulative distribution function at 8 from the cumulative distribution function at 3:

$P(4\;to\;8\;Heads) = P(0\;to\;8\;Heads) - P(0\;to\;3\;Heads)$

In python, we use the following code:

import scipy.stats as stats

print(stats.binom.cdf(8, 10, 0.5) - stats.binom.cdf(3, 10, 0.5))

Output:

# 0.81738

To calculate the probability of observing more than 6 heads from 10 fair coin flips we subtract the value of the cumulative distribution function at 6 from 1. Mathematically, this looks like the following:

$P(more\; than\; 6\; heads) = 1 - P(6\; or\; fewer\; heads)$

Note that “more than 6 heads” does not include 6. In python, we would calculate this probability using the following code:

import scipy.stats as stats
print(1 - stats.binom.cdf(6, 10, 0.5))

Output:

# 0.171875