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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  Normalization & z-score example: werewolf transform

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

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