🎯 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:

Fixed number of trials → n
Each trial is independent
Only two outcomes per trial
Probability of success p is constant

Key Terms & Notations

Symbol	Meaning
`n`	Number of trials
`x`	Number of successes
`p`	Probability of success
`q`	Probability of failure (`q = 1 − p`)
`X`	Random variable

Binomial Probability Mass Function (PMF)

Formula

P(X = x) = ⁿCₓ pˣ qⁿ−ˣ

Where:

ⁿCₓ = n! / x!(n−x)!

Understanding the Formula

ⁿCₓ → Number of ways to choose x successes
pˣ → Probability of x successes
qⁿ−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) = ³C₂ (1/2)² (1/2)¹

Step 3: Calculate components

Component	Calculation	Value
Combination	³C₂	3
p²	(1/2)²	1/4
q¹	(1/2)	1/2

Step 4: Final Probability

P(X = 2) = 3 × 1/4 × 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	Expression	Value
Combination	⁵C₁	5
p¹	0.1	0.1
q⁴	(0.9)⁴	0.6561

Step 3: Final Answer

P(X = 1) = 5 × 0.1 × 0.6561 = 0.32805

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