Binomial Distribution Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Binomial Distribution.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The probability distribution of the number of successes in $n$ independent yes/no trials, each with probability $p$ .

Flip a biased coin $n$ times—how many heads? The binomial distribution gives the probability of each count.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The binomial distribution gives the probability of getting exactly $k$ successes in $n$ independent yes/no trials, each with the same chance $p$ .

Common stuck point: The procedure for binomial distribution is the easy part; the trap is applying it when trials aren't independent or $p$ changes. Asking "Are there a fixed number of independent trials, each success/failure with the same probability, and I want a count of successes?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are there a fixed number of independent trials, each success/failure with the same probability, and I want a count of successes?

Worked Examples

Example 1

medium

A fair coin is flipped

8

times. What is the probability of getting exactly

5

heads?

Answer

P(X = 5) = \frac{7}{32} \approx 0.219

First step

Use the binomial formula:

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

, where

n = 8

k = 5

p = 0.5

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Example 2

hard

A multiple-choice quiz has

10

questions, each with

4

choices. If a student guesses randomly, what is the probability of getting at least

2

correct?

Example 3

medium

A pharmacy fills 12 prescriptions; each has a 5% error rate independently. Find

P(\text{no errors})

Example 4

medium

A vaccine is 92% effective. In a group of 25 vaccinated people exposed to the pathogen, find the probability that more than 24 (i.e., all 25) are protected.

Example 5

hard

In a town, 60% of registered voters favor candidate A. A random sample of 5 voters is polled. Find the probability that at least 4 favor A.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

A basketball player has a

70\%

free-throw success rate. In

6

attempts, what is the probability of making exactly

4

Example 2

medium

A student guesses on

7

true-or-false questions. What is the probability of getting at least

6

correct?

Example 3

easy

A fair coin is flipped 3 times. How many ways give exactly 2 heads? Use

\binom{3}{2}

Example 4

easy

For

n=4

p=0.5

, write

P(X=k)

for a fair coin. Find

P(X=4)

Example 5

easy

A binomial has

n=10

p=0.3

. What is the expected number of successes?

Example 6

easy

Identify whether 'number of heads in 5 flips' is binomial. State

n

and

p

Example 7

easy

P(X=0)

for

n=3

p=0.2

. Compute.

Example 8

easy

A binomial coefficient

\binom{5}{2}

equals what?

Example 9

easy

For

n=6

p=0.5

, is the distribution symmetric? Why?

Example 10

easy

P(X=1)

for

n=2

p=0.5

. Compute.

Example 11

medium

A quiz has 5 true/false questions answered by guessing.

P(\text{exactly 3 correct})=?

Example 12

medium

n=4

p=0.3

. Find

P(X=2)

Example 13

medium

n=5

p=0.2

. Find

P(X\ge 1)

Example 14

medium

n=8

p=0.25

. Find the variance of the binomial.

Example 15

medium

A factory: 10% of items defective. In a box of 5,

P(\text{exactly 1 defective})=?

Example 16

medium

Why is drawing 3 cards (no replacement) and counting aces NOT binomial?

Example 17

medium

n=3

p=0.5

. Find

P(X\le 1)

Example 18

medium

A binomial with

n=20

p=0.5

has what mean and standard deviation?

Example 19

medium

n=7

p=0.5

. Find

P(X=7)

(all successes).

Example 20

challenge

n=4

p=0.5

. Find

P(X=2 \mid X\ge 1)

Example 21

challenge

For what

p

is the binomial variance

np(1-p)

maximized, for fixed

n

Example 22

challenge

A student needs

\ge 4

correct of 5 (each

p=0.8

) to pass. Find

P(\text{pass})

Example 23

easy

For

n=5

p=0.4

, write the formula for

P(X=2)

Example 24

easy

Compute

P(X=2)

for

n=5

p=0.4

Example 25

easy

For

n=3

p=0.5

, find

P(X\ge 2)

Example 26

medium

A free-throw shooter has

p=0.6

. In

n=10

attempts, find

E[X]

and

\sigma_X

Example 27

medium

For

n=6

p=0.3

, find

P(X\le 1)

Example 28

medium

A multiple-choice test has 20 questions with 4 choices each, all guessed. Find

E[X]

and

\sigma_X

Example 29

medium

For

n=5

p=0.4

, find

P(X=0)

Example 30

medium

For

n=8

p=0.5

, find the mode of

X

Example 31

medium

For

n=4

p=0.5

, find

P(X\ge 3)

Example 32

medium

For

n=5

p=0.6

, find

P(X=3)

Example 33

medium

Show: if

X\sim \text{Bin}(n,p)

, then

n-X\sim \text{Bin}(n,1-p)

Example 34

hard

For

n=10

p=0.2

, use the binomial pmf to compute

P(X=3)

Example 35

hard

A factory ships boxes of 20 widgets,

p=0.03

defective per widget. Find

P(\text{at least one defective})

Example 36

hard

A test of

n=20

items must pass if defects are

\le 1

. Defect rate

p=0.05

. Find

P(\text{pass})

Example 37

hard

For

X\sim \text{Bin}(n,p)

, derive

E[X(X-1)]

Example 38

hard

For

n=12

p=0.25

, find

P(X=3)

Example 39

challenge

What's the smallest

n

for which a fair coin has

P(X\ge 1)\ge 0.99

? (

X\sim\text{Bin}(n,0.5)

Example 40

challenge

For

n=100

p=0.5

, use the normal approximation with continuity correction to estimate

P(X\ge 55)

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Related Concepts

Binomial Coefficient Probability Independent Events Normal Distribution

Background Knowledge

These ideas may be useful before you work through the harder examples.

binomial coefficientprobabilityindependent events