Skip to content

Binomial distribution

⭐ Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success.

Think of it as the math of “How many times will this work out of n tries?”

🎯 When to Use It

Use the binomial distribution when:

  • You have a fixed number of trials n
  • Each trial has two outcomes: success or failure
  • Probability of success p is constant
  • Trials are independent
  • You’re counting how many successes occur

Examples:

  • Number of heads in 10 coin flips
  • Number of correct answers on a 20‑question multiple‑choice test (guessing)
  • Number of customers who buy something out of 50
  • Number of defective items in a batch of 100

📌 Probability Formula

If X is the number of successes in n trials:

P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}

Where:

  • k = number of successes
  • p = probability of success
  • \binom{n}{k} = number of ways to choose which trials are successes

🧠 Expected Value and Variance

E(X) = np

\text{Var}(X) = np(1 - p)

⭐ Examples

Example 1: Coin Flips

Flip a fair coin 5 times.
Let X = number of heads.
Here:

  • n = 5
  • p = 0.5

Probability of getting exactly 3 heads:

P(X = 3) = \binom{5}{3}(0.5)^3(0.5)^2
= 10 \cdot 0.125 \cdot 0.25 = 0.3125

Example 2: Multiple‑Choice Guessing

A student guesses on 10 multiple‑choice questions, each with 4 options.

  • n = 10
  • p = 0.25 (chance of guessing correctly)

Probability of getting exactly 4 correct:

P(X = 4) = \binom{10}{4}(0.25)^4(0.75)^6

Expected number correct:

E(X) = np = 10(0.25) = 2.5

Example 3: Manufacturing Defects

A factory produces items with a 3% defect rate.

Let X = number of defective items in a batch of 100.

  • n = 100
  • p = 0.03

Probability exactly 2 items are defective:

P(X = 2) = \binom{100}{2}(0.03)^2(0.97)^{98}

Expected number of defects:

E(X) = 100(0.03) = 3

Example 4: Basketball Free Throws

A player makes free throws with probability p = 0.8.
She takes 15 shots.

Let X = number of made shots.

Probability she makes at least 12:

P(X \ge 12) = \sum_{k=12}^{15} \binom{15}{k}(0.8)^k(0.2)^{15-k}

Expected makes:

E(X) = 15(0.8) = 12

🎨 Intuition

The binomial distribution is like a scoreboard:

  • You try something n times
  • Each try has the same chance of success
  • You count how many times it works
See also  expectation is linear

It’s the backbone of probability in real‑world settings.

Discover more from Knowledge sparks

Subscribe to get the latest posts sent to your email.

Leave a Reply

error: Content is protected !!