Binomial Distribution

Last Updated: Jan 2026

Binomial Distribution is a discrete probability distribution used when an experiment is repeated a fixed number of times, and each trial has only two possible outcomes.

These outcomes are usually called:

• Success
• Failure

🗣 Hinglish Tip: Binomial = do result (success / failure) + fixed trials

When to Use Binomial Distribution?

Binomial distribution is applicable only if all conditions are satisfied:

1. Fixed number of trials → n
2. Each trial is independent
3. Only two outcomes per trial
4. Probability of success p is constant

Key Terms & Notations

Binomial Probability Mass Function (PMF)

Formula

$P(X = x) = \binom{n}{x} p^x q^{n-x}$

Where:

$\binom{n}{x} = \frac{n!}{x!(n-x)!}$

Understanding the Formula

• $\binom{n}{x}$ → Number of ways to choose x successes
• $p^x$ → Probability of x successes
• $q^{n-x}$ → Probability of remaining failures

Example 1: Coin Toss

A fair coin is tossed 3 times. Find the probability of getting exactly 2 heads.

Step 1: Identify parameters

n = 3
x = 2
p = 1/2
q = 1 - 1/2 = 1/2

Step 2: Apply formula

$P(X = 2) = \binom{3}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^1$

Step 3: Calculate components

Step 4: Final Probability

$P(X = 2) = 3 \times 1/4 \times 1/2 = 3/8$

Example 2: Defective Items

A factory produces items with 10% defect rate. If 5 items are selected randomly, find probability that exactly 1 is defective.

Step 1: Values

n = 5
x = 1
p = 0.1
q = 0.9

Step 2: Calculation Table

Step 3: Final Answer

$P(X = 1) = 5 \times 0.1 \times  0.6561 = 0.32805$

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