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Class 12 Statistics Notes · GSEB

Binomial Distribution

Binomial Distribution — apply the formula P(X = x) = nCx · p^x · q^(n−x) to compute probabilities in repeated Bernoulli trials. GSEB Class 12 Commerce Statistics notes with interactive calculator.

Last updated: 22 Sep 2026

Notes

The Binomial Distribution

Binomial Distribution
The binomial distribution gives the probability of getting exactly x successes in n independent Bernoulli trials, each with the same probability of success p. It is written as X ~ B(n, p).

Binomial Probability Mass Function

P(X=x)=nCx⋅px⋅qn−xP(X = x) = {}^nC_x \cdot p^x \cdot q^{n-x}
The name "binomial" comes from the binomial theorem: (p + q)^n = Σ nCx · p^x · q^(n−x). Since p + q = 1, (p + q)^n = 1 = Σ P(X = x) — which is exactly our condition that probabilities sum to 1.

Understanding the Formula

{n}C{x}

Number of ways to choose which x trials are successes out of n total trials.

C = n! / (x! × (n−x)!)

p^x

Probability of x successes. Each success contributes a factor of p.

q^{n-x}

Probability of (n − x) failures. Each failure contributes a factor of q = 1 − p.

Try It: Interactive Calculator

Adjust n and p with the sliders. Click any probability bar to see the full calculation.

Binomial Distribution Calculator — X ~ B(n, p)
110
01
X ~ B(3, 0.50)|μ = np = 1.50|σ² = npq = 0.75
x0123
P(X = x)0.12500.37500.37500.1250
0.125
0
0.375
1
0.375
2
0.125
3

nCx(0)

1

ΣP(x)

1.0000

Mean μ

1.50

Variance σ²

0.75

Solved Examples

Solved Example

Problem

Three coins are tossed. Find P(X = 2) where X = number of heads.

Solution

P(X = 2) = 0.375

Solved Example

Problem

A die is rolled 4 times. Find P(getting exactly 2 sixes).

Solution

P(X = 2) = 150/1296 ≈ 0.1157

Solved Example

Problem

80% of bulbs are non-defective. Find the probability that in a batch of 5, exactly 4 are non-defective.

Solution

P(X = 4) = 0.4096

Conditions for Binomial Distribution

All four conditions must hold

  1. Fixed number of trials (n is known in advance)
  2. Two outcomes only — success or failure
  3. Independent trials — one result doesn't affect another
  4. Constant probability — p stays the same for every trial

If any condition fails, the binomial distribution does NOT apply.

Key Takeaways

Key Takeaways

  • P(X = x) = ⁿCₓ · pˣ · qⁿ⁻ˣ — the binomial probability formula.
  • X ~ B(n, p) means X follows a binomial distribution with n trials and success probability p.
  • nCx counts the number of ways to arrange x successes in n trials.
  • X can take values 0, 1, 2, ..., n. ΣP(X = x) for x = 0 to n always equals 1.
  • Four conditions: fixed n, two outcomes, independent trials, constant p.

Practice

  1. Five coins are tossed. Find P(X = 3) where X = number of heads.
  2. A bag has 60% red balls. 3 balls are drawn with replacement. Find P(all 3 are red).
  3. Why does the binomial distribution require replacement (or a very large population)?

Explore Further