8.2 Binomial Probability

If an experiment consists of $n$ identical and independent trials, with probability of success $p$ (remains the same for each trial), then the random variable $X$ defined as the number of successes in $n$ trials is a binomial random variable and the probability of $x$ successes is given by:

$$P(X=x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^k(1-p{)}^{n-x}$$

where $x=0,1,2,3,...,n$

The syntax for $P(X=x)$, given n and p is

from scipy import stats
stats.binom.pmf(x,n,p)

The syntax for cumulative probability $P(X\le )$, given n and p is

from scipy import stats
stats.binom.cdf(x,n,p)

The syntax for mean and standard deviation is given by:

from scipy import stats
stats.binom.mean(n,p)
stats.binom.std(n,p) 

Example: Find $P(X=10)$ if $p=0.6,n=12$

$$P(x=10)=\left(\genfrac{}{}{0ex}{}{12}{10}\right)6^{10}(1-0.6{)}^{12-10}$$

Here x=10, n=12 and p=0.6.

from scipy import stats
stats.binom.pmf(10, 12, 0.6)

0.06385228185599987

Example: For a Binomial distribution with 12 number of trials and probability of success 0.6, find the cumulative probabilities $$P(x\le 2)={\Sigma }_1^{\mathrm{.6}}\left(\genfrac{}{}{0ex}{}{12}{2}\right){\mathrm{.6}}^2(1-\mathrm{.6}{)}^{12-2}$$. Here x=2, p=0.6 and n=12.

from scipy import stats
stats.binom.cdf(k=2, n=12, p=0.6)  # same as stats.binom.cdf(2, 12, 0.6)

0.00281018368

Interpretation: $P(X\le 2)=0.00281018368$

Example: A manufacturing machine has a 5% defect rate. Write a scipy.stats program to compute the following:

• If 6 items are chosen at random, what is the probability of exactly 2 defects.

from scipy import stats

stats.binom.pmf(2,6,0.05)

0.030543984375000013

Interpretation: The probability of exactly two out of six defects is 0.030543984375000013

Example: A manufacturing machine has a 5% defect rate. Write a scipy.stats program to compute the following:

• If 6 items are chosen at random, what is the probability that 3 or less will have a defect?

from scipy import stats
stats.binom.cdf(3,6,0.05)

0.99991359375

Interpretation: The probability of 3 or less defects is 0.99991359375

Example: A manufacturing machine has a 5% defect rate. Write a scipy.stats program to compute the following:

• If 6 items are chosen at random, what is the probability that at least two defects?

from scipy import stats
1-stats.binom.cdf(1,6,0.05)

0.03277382812500007

Interpretation: The probability of at least two defects is 0.03277382812500007

Example: 20% of cars fail emission inspections in IL. If 15 cars are randomly selected, what is the probability that exactly three fail.

Here x=3, p=.2 and n=15

Watch demo video

from scipy import stats
stats.binom.pmf(3,15,.2)

0.2501388953190411

Interpretation: $P(X=3)=0.00281018368$

Example: 20% of cars fail emission inspections in IL. If 15 cars are randomly selected, what is the probability that at most five fail?

Here x=5, p=.15 and n=15 (Cumulative)

Watch demo video

from scipy import stats
stats.binom.cdf(5,15,.2)

0.938948570382336

Interpretation: $$P(X\le 5)=0.938948570382336$$

Example: 20% of cars fail emission inspections in IL. If 15 cars are randomly selected, what is the probability that at least five fail?

$$P(X\ge 5)=1-P(X\le 4)$$

from scipy import stats
1 - stats.binom.cdf(4,15,.2)

0.16423372393676794

Interpretation: $$P(X\ge 5)=0.16423372393676794$$

Example: 20% of cars fail emission inspections in IL.15 cars are randomly selected from IL. Make a probability distribution table for each outcome.

from scipy import stats
import numpy as np
x = np.arange(0,15+1)  #generates [0,1,2,..,15]
y = stats.binom.pmf(x,15,.2)

prob = zip(x,y)
list(prob)

[(0, 0.035184372088831996),
 (1, 0.13194139533312002),
 (2, 0.23089744183296135),
 (3, 0.2501388953190411),
 (4, 0.18760417148928044),
 (5, 0.10318229431910414),
 (6, 0.04299262263296016),
 (7, 0.013819057274880048),
 (8, 0.0034547643187200112),
 (9, 0.000671759728640003),
 (10, 0.00010076395929600008),
 (11, 1.1450449920000038e-05),
 (12, 9.542041600000053e-07),
 (13, 5.505024000000039e-08),
 (14, 1.9660800000000028e-09),
 (15, 3.276800000000002e-11)]

Interpretation:

Binomial Plot

Use the function binomplot below to plot a binomial distribution with probability p and number of trials n.

# binomial graph with n and p
def binomplot(n,p):
  from scipy import stats
  import numpy as np
  import matplotlib.pyplot as plt

  x = np.arange(0,n+1)
  y = stats.binom.pmf(x,n,p)
  plt.bar(x,y,width=0.2)

  plt.suptitle("Binomial Plot")
  plt.xlabel("x values")
  plt.ylabel("probability")

  return plt.show()

Example: plot a binomial distribution with n=10 and p=0.6

binomplot(10,.6)   # run the binomplot function above first

Example: Graph a Binomial Probability Distribution with n=10 and p=.3.

binomplot(10,.3)    # run the binomplot function above first

Adapted from Python for Introductory Statistics, by Simon Aman (Truman College, City Colleges of Chicago), licensed under CC BY 4.0. Changes were made: reformatted as an accessible XYZ web edition with live in-browser code cells. License: CC-BY-4.0.