Binomial Distribution in NumPy

The Binomial Distribution models the number of successes in a fixed number of independent trials where each trial has only two outcomes: success or failure. In NumPy, we use the numpy.random.binomial() method to generate values that follow this distribution. It is commonly used in coin flips, defect detection, surveys, and probability experiments.

Example: Here, we generate one binomial random value using 10 trials and a 0.5 probability of success.

import numpy as np
x = np.random.binomial(n=10, p=0.5)
print(x)

4

Explanation: np.random.binomial(n=10, p=0.5) simulates 10 yes/no events and returns how many times success occurred.

Syntax

numpy.random.binomial(n, p, size=None)

Parameters:

• n: Number of trials
• p: Probability of success in each trial
• size: Shape of output array

Examples

Example 1: In this example, we generate 5 binomial random numbers using 10 trials and 0.5 probability.

import numpy as np
arr = np.random.binomial(n=10, p=0.5, size=5)
print(arr)

[7 2 5 4 7]

Explanation: np.random.binomial(..., size=5) returns an array of 5 simulated outcomes.

Example 2: Here, we simulate 8 trials with different success probability (p = 0.3).

import numpy as np
x = np.random.binomial(8, 0.3, size=4)
print(x)

[4 4 4 4]

Explanation: np.random.binomial(8, 0.3) generates values where success occurs with 30% probability.

Example 3: In this example, we generate a 2×3 matrix of binomial outcomes.

import numpy as np
m = np.random.binomial(12, 0.6, size=(2, 3))
print(m)

[[8 7 9]
 [6 8 5]]

Explanation: size=(2,3) creates a 2D array where each entry is a binomial random value.

Visualizing the Binomial Distribution

Visualizing the generated numbers helps in understanding their behavior. Below is an example of plotting a histogram of random numbers generated using numpy.random.binomial.

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import binom

n = 10
p = 0.5
size = 1000

data = np.random.binomial(n, p, size)

plt.hist(data, bins=np.arange(-0.5, n+1.5, 1), density=True, edgecolor='black', alpha=0.7, label='Histogram')

x = np.arange(0, n+1)
pmf = binom.pmf(x, n, p)

plt.scatter(x, pmf, color='red', label='Theoretical PMF')
plt.vlines(x, 0, pmf, colors='red', linestyles='dashed')

plt.title("Binomial Distribution (n=10, p=0.5)")
plt.xlabel("Number of Successes")
plt.ylabel("Probability")
plt.legend()
plt.grid(True)
plt.show()

Output

Explanation:

• np.random.binomial(n, p, size) generates 1000 simulated outcomes.
• plt.hist(..., density=True) shows the frequency distribution of these values.
• binom.pmf(x, n, p) computes the theoretical probability for each possible success count.
• Red dots and dashed lines show the true Binomial PMF for comparison.