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Statistics

Binomial Probability

Lesson

A binomial setting is exactly what it sounds like: nn independent trials, each with the same probability ppof “success.” Examples:

  • Flip a coin 5 times. How many heads?
  • Take 10 free throws at 70%. How many made?
  • Guess on 4 multiple-choice questions. How many correct?

The probability of exactly kk successes in nn trials:

P(X=k)=(nk) pk (1−p)n−kP(X = k) = \binom{n}{k}\,p^k\,(1-p)^{n-k}

Three pieces:

  • (nk)\binom{n}{k} — the number of orderings of kk successes among nn trials.
  • pkp^k — the probability that all kk successful trials succeed.
  • (1−p)n−k(1-p)^{n-k} — the probability that the remaining n−kn - k trials all fail.

Worked example 1

Flip a fair coin 4 times. What’s the probability of exactly 2 heads?

n=4, k=2, p=12n = 4,\ k = 2,\ p = \tfrac{1}{2}.

P(X=2)=(42)(12) ⁣2 ⁣(12) ⁣2=6⋅116=616=38\begin{aligned} P(X = 2) &= \binom{4}{2}\left(\tfrac{1}{2}\right)^{\!2}\!\left(\tfrac{1}{2}\right)^{\!2} \\ &= 6 \cdot \tfrac{1}{16} \\ &= \tfrac{6}{16} = \tfrac{3}{8} \end{aligned}

Worked example 2

Roll 3 dice. What’s the probability of exactly one 6?

Each die: success probability p=1/6p = 1/6, failure 5/65/6. n=3, k=1n = 3,\ k = 1.

P(X=1)=(31)(16) ⁣1 ⁣(56) ⁣2=3⋅16⋅2536=75216=2572\begin{aligned} P(X = 1) &= \binom{3}{1}\left(\tfrac{1}{6}\right)^{\!1}\!\left(\tfrac{5}{6}\right)^{\!2} \\ &= 3 \cdot \tfrac{1}{6} \cdot \tfrac{25}{36} \\ &= \tfrac{75}{216} = \tfrac{25}{72} \end{aligned}

How to type your answer

A fraction in lowest terms or a decimal. Examples: 3/8, 25/72, 5/16, 1/256.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

4 coin flips. P(exactly 2 heads)\text{4 coin flips. } P(\text{exactly 2 heads})

Problem 2

5 coin flips. P(exactly 3 heads)\text{5 coin flips. } P(\text{exactly 3 heads})

Problem 3

3 coin flips. P(exactly 0 heads)\text{3 coin flips. } P(\text{exactly 0 heads})

Problem 4

3 coin flips. P(exactly 3 heads)\text{3 coin flips. } P(\text{exactly 3 heads})

Practice

Standard problems matching the lesson.

Problem 5

5 coin flips. P(0 heads)\text{5 coin flips. } P(0 \text{ heads})

Problem 6

5 coin flips. P(exactly 1 head)\text{5 coin flips. } P(\text{exactly 1 head})

Problem 7

5 coin flips. P(5 heads)\text{5 coin flips. } P(5 \text{ heads})

Problem 8

6 coin flips. P(exactly 3 heads)\text{6 coin flips. } P(\text{exactly 3 heads})

Problem 9

3 dice. P(exactly 1 six)\text{3 dice. } P(\text{exactly 1 six})

Problem 10

3 dice. P(0 sixes)\text{3 dice. } P(\text{0 sixes})

Problem 11

3 dice. P(exactly 2 sixes)\text{3 dice. } P(\text{exactly 2 sixes})

Problem 12

3 dice. P(all 3 sixes)\text{3 dice. } P(\text{all 3 sixes})

Problem 13

50% free-throw shooter, 4 shots. P(exactly 3 made)\text{50\% free-throw shooter, 4 shots. } P(\text{exactly 3 made})

Problem 14

50% free-throw shooter, 4 shots. P(all 4 made)\text{50\% free-throw shooter, 4 shots. } P(\text{all 4 made})

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

4 multiple-choice (4 options), all guessed. P(exactly 1 correct)\text{4 multiple-choice (4 options), all guessed. } P(\text{exactly 1 correct})

Problem 16

Same setup. P(all 4 wrong)\text{Same setup. } P(\text{all 4 wrong})

Problem 17

Same setup. P(all 4 correct)\text{Same setup. } P(\text{all 4 correct})

Problem 18

6 coin flips. P(exactly 4 heads)\text{6 coin flips. } P(\text{exactly 4 heads})

Problem 19

Drug works 4/5 of the time, 5 patients. P(all 5 cured)\text{Drug works } 4/5 \text{ of the time, 5 patients. } P(\text{all 5 cured})

Problem 20

Same drug, 5 patients. P(exactly 4 cured)\text{Same drug, 5 patients. } P(\text{exactly 4 cured})

Problem 21

4 dice. P(no 6s)\text{4 dice. } P(\text{no 6s})

Problem 22

P(H)=1/3. 3 flips. P(exactly 2 heads)P(H) = 1/3.\ \text{3 flips. } P(\text{exactly 2 heads})