Binomial Distribution

🎲 Binomial Distribution in Python

The Binomial Distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.
It’s widely used in coin tosses, quality control, and yes/no experiments.

1. Characteristics of Binomial Distribution

n → number of trials
p → probability of success in each trial
size → number of experiments / samples
Values range from 0 to n

Probability Mass Function (PMF):

$\binom{n}{k} p^k (1-p)^{n-k}$

Where:

$k$ = number of successes
$(nk)\binom{n}{k}$ = combinations of n choose k

2. Generate Binomial Data Using NumPy

import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns# Parameters
n = 10 # number of trials
p = 0.5 # probability of success
size = 1000 # number of experiments

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

print(data[:10]) # first 10 outcomes

import numpy as np

import matplotlib.pyplot as plt

import seaborn as sns# Parameters

n = 10 # number of trials

p = 0.5 # probability of success

size = 1000 # number of experiments

# Generate random data

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

print(data[:10]) # first 10 outcomes

Output (example):

3. Visualize Binomial Distribution

Peaks around expected value: E[X] = n*p
Histogram shows discrete outcomes

4. Change Probability or Number of Trials

5. Compare Two Probabilities

data1 = np.random.binomial(n=10, p=0.3, size=1000)
data2 = np.random.binomial(n=10, p=0.7, size=1000)sns.histplot(data1, bins=range(12), color='red', label='p=0.3', kde=False)
sns.histplot(data2, bins=range(12), color='green', label='p=0.7', kde=False)
plt.legend()
plt.title('Binomial Distribution Comparison')
plt.show()

data1 = np.random.binomial(n=10, p=0.3, size=1000)

data2 = np.random.binomial(n=10, p=0.7, size=1000)sns.histplot(data1, bins=range(12), color='red', label='p=0.3', kde=False)

sns.histplot(data2, bins=range(12), color='green', label='p=0.7', kde=False)

plt.legend()

plt.title('Binomial Distribution Comparison')

plt.show()

Lower probability → peak closer to 0
Higher probability → peak closer to n

6. Compute Mean and Variance

Theoretical values:

Mean = $n * p$
Variance = $n * p * (1 - p)$

🎯 Practice Exercise

Simulate 50 coin flips (n=1) 1000 times and plot histogram.
Simulate quality check: 20 items, success probability 0.8, 1000 experiments.
Compare p=0.2 vs p=0.8 for 10 trials.

Binomial Distribution