Statistics

Binomial Probability

Lesson

A binomial setting is exactly what it sounds like: $n$ independent trials, each with the same probability $p$ of “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 $k$ successes in $n$ trials:

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

Three pieces:

$\binom{n}{k}$ — the number of orderings of $k$ successes among $n$ trials.
$p^k$ — the probability that all $k$ successful trials succeed.
$(1-p)^{n-k}$ — the probability that the remaining $n - k$ trials all fail.

Worked example 1

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

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

\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 $p = 1/6$ , failure $5/6$ . $n = 3,\ k = 1$ .

\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

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

Problem 2

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

Problem 3

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

Problem 4

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

Practice

Standard problems matching the lesson.

Problem 5

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

Problem 6

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

Problem 7

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

Problem 8

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

Problem 9

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

Problem 10

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

Problem 11

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

Problem 12

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

Problem 13

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

Problem 14

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

Challenge

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

Problem 15

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

Problem 16

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

Problem 17

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

Problem 18

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

Problem 19

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

Problem 20

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

Problem 21

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

Problem 22

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

Practice

Standard problems matching the lesson.

Problem 23

Flip a fair coin 4 times. P(exactly 2 heads)?

Problem 24

Flip a fair coin 5 times. P(exactly 3 heads)?

Challenge

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

Problem 25

Roll a fair die 3 times. A success = rolling a 6. P(exactly 1 success)?

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