🎲 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):

P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn​)pk(1−p)n−k

Where:

• $k$ = number of successes

• (nk)\binom{n}{k}(kn​) = combinations of n choose k

✅ 2. Generate Binomial Data Using NumPy

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

• Lower probability → peak closer to 0

• Higher probability → peak closer to n

✅ 6. Compute Mean and Variance

Theoretical values:

• Mean = $n\ast p$

• Variance = $n\ast p\ast (1-p)$

🎯 Practice Exercise

1. Simulate 50 coin flips (n=1) 1000 times and plot histogram.

2. Simulate quality check: 20 items, success probability 0.8, 1000 experiments.

3. Compare p=0.2 vs p=0.8 for 10 trials.