Binomial Distribution

 Understanding the Binomial Distribution

The Binomial Distribution is a discrete probability distribution that models the number of successes in a fixed number of independent experiments, where each experiment has only two possible outcomes: success or failure.

 Key Characteristics

Discrete Distribution:

• Binomial Distribution applies to scenarios with distinct outcomes. For example, flipping a coin yields either heads or tails—there's no in-between.

Binary Outcomes:

• Each trial has exactly two outcomes.

Parameters:

• n → Total number of trials (experiments).
• p → Probability of success in each trial (e.g., 0.5 for a fair coin).
• size → The number of random samples to generate.

Program: 

Simulating Coin Tosses

Let’s simulate 10 coin tosses, repeated 10 times:

from numpy import random

x = random.binomial(n=10, p=0.5, size=10)

print(x)

This will generate 10 data points representing the number of successful outcomes (e.g., number of heads) in each set of 10 coin tosses.

Output:

[4 5 6 3 6 6 3 7 4 2]

Visualizing the Binomial Distribution

Here’s how you can visualize the distribution using Python:

Program:

from numpy import random

import matplotlib.pyplot as plt

import seaborn as sns

sns.displot(random.binomial(n=10, p=0.5, size=1000))

plt.show()

This plot shows how the outcomes are distributed after repeating the experiment 1,000 times.

Difference Between Normal and Binomial Distribution

The primary difference between the normal and binomial distributions lies in their nature:

Normal distribution is continuous.

Binomial distribution is discrete.

However, with a large enough number of trials, the binomial distribution tends to approximate the normal distribution, given appropriate values for the mean (loc) and standard deviation (scale).

Program:

from numpy import random

import matplotlib.pyplot as plt

import seaborn as sns

# Generate data for normal and binomial distributions

data = {

    "Normal": random.normal(loc=50, scale=5, size=1000),

    "Binomial": random.binomial(n=100, p=0.5, size=1000)

}

# Plot the distributions

sns.displot(data, kind="kde")

plt.show()

This example demonstrates how a binomial distribution can resemble a normal distribution when the sample size is large.