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Learn Index · Probability and Combinatorics Grade 11–12

Binomial probability

The chance of exactly k successes in n independent trials.

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Two outcomes, repeated

A Bernoulli trial has exactly two outcomes: success or failure. A binomial experiment repeats such trials.

Conditions (BINS):

  • Binary: two outcomes each trial
  • Independent trials
  • Number of trials fixed
  • Same probability of success p each time

The formula

P(X = k) = nCk × pᵏ × (1 − p)ⁿ⁻ᵏ

  • n = number of trials
  • k = number of successes
  • p = probability of success
  • nCk counts the ways to arrange the successes

Example

A basketball player makes 70% of free throws. In 5 shots, what’s P(exactly 3 made)?

  • n = 5, k = 3, p = 0.7
  • 5C3 = 10
  • P = 10 × 0.7³ × 0.3² = 10 × 0.343 × 0.09 = 0.3087

Why nCk?

Making 3 of 5 could happen as MMMXX, MXMXM, XXMMM, and so on. Each has probability 0.7³ × 0.3², and there are 5C3 = 10 such orders.

"At least" and "at most"

At least 1 success: 1 − P(0)

Example: P(at least 1 six in 4 rolls) = 1 − (5/6)⁴ ≈ 0.518

At most 2: P(0) + P(1) + P(2)

The binomial distribution

Listing P(X = k) for all k gives the binomial distribution.

Example: n = 4 coin flips, p = 0.5

Heads k 0 1 2 3 4
P 1/16 4/16 6/16 4/16 1/16

The numerators 1, 4, 6, 4, 1 come from Pascal’s triangle.

Mean and spread

  • Expected value (mean): μ = np
  • Standard deviation: σ = √[np(1 − p)]

Example: 100 coin flips → μ = 50 heads, σ = 5.

Technology

Spreadsheets: =BINOM.DIST(k, n, p, FALSE) for exactly k; TRUE for at most k. Calculators have binompdf and binomcdf.

Not binomial

  • Drawing cards without replacement (p changes)
  • Counting until the first success (n isn’t fixed)

Worked examples

Example 1: P(exactly 2 heads in 3 flips)? 3C2 × 0.5³ = 3/8.

Example 2: A test has 10 multiple-choice questions with 4 options each. Guessing, P(exactly 5 correct)? 10C5 × 0.25⁵ × 0.75⁵ ≈ 0.058.

Example 3: 20% of bulbs are defective. Expected defectives in 50? 10.

Common mistakes

  • Forgetting nCk.
  • Using the formula when trials aren’t independent.
  • Mixing up "at least" and "exactly".

Try it yourself

  1. P(exactly 4 heads in 6 flips)?
  2. p = 0.2, n = 5. P(0 successes)?
  3. P(at least 1 success) for question 2?
  4. n = 200, p = 0.3. Mean and standard deviation?

Answers: 1) 15/64 ≈ 0.234 2) 0.8⁵ ≈ 0.328 3) ≈ 0.672 4) 60 and ≈ 6.48

Quick recap

  • Binomial probability: P(X = k) = nCk pᵏ (1 − p)ⁿ⁻ᵏ.
  • Requires fixed, independent Bernoulli trials with constant p.
  • Mean (expected value) np; SD √[np(1 − p)].