Practice Binomial Distribution in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The probability distribution of the number of successes in nn independent yes/no trials, each with probability pp.

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

Showing a random 20 of 50 problems.

Example 1

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 2

medium
A basketball player has a 70%70\% free-throw success rate. In 66 attempts, what is the probability of making exactly 44?

Example 3

hard
A multiple-choice quiz has 1010 questions, each with 44 choices. If a student guesses randomly, what is the probability of getting at least 22 correct?

Example 4

medium
A binomial with n=20n=20, p=0.5p=0.5 has what mean and standard deviation?

Example 5

challenge
For n=100n=100, p=0.5p=0.5, use the normal approximation with continuity correction to estimate P(X≥55)P(X\ge 55).

Example 6

easy
For n=10n=10, p=0.5p=0.5, what is Var(X)\text{Var}(X)?

Example 7

medium
A fair coin is flipped 88 times. What is the probability of getting exactly 55 heads?

Example 8

medium
For n=5n=5, p=0.6p=0.6, find P(X=3)P(X=3).

Example 9

medium
For n=6n=6, p=0.3p=0.3, find P(X≤1)P(X\le 1).

Example 10

challenge
For what pp is the binomial variance np(1−p)np(1-p) maximized, for fixed nn?

Example 11

medium
n=8n=8, p=0.25p=0.25. Find the variance of the binomial.

Example 12

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For n=8n=8, p=0.5p=0.5, find the mode of XX.

Example 13

challenge
A student needs ≥4\ge 4 correct of 5 (each p=0.8p=0.8) to pass. Find P(pass)P(\text{pass}).

Example 14

easy
For n=6n=6, p=0.5p=0.5, is the distribution symmetric? Why?

Example 15

medium
Show: if X∼Bin(n,p)X\sim \text{Bin}(n,p), then n−X∼Bin(n,1−p)n-X\sim \text{Bin}(n,1-p).

Example 16

easy
What are the two parameters of a binomial distribution?

Example 17

hard
A test of n=20n=20 items must pass if defects are ≤1\le 1. Defect rate p=0.05p=0.05. Find P(pass)P(\text{pass}).

Example 18

hard
For n=12n=12, p=0.25p=0.25, find P(X=3)P(X=3).

Example 19

easy
Identify whether 'number of heads in 5 flips' is binomial. State nn and pp.

Example 20

medium
For n=4n=4, p=0.5p=0.5, find P(X≥3)P(X\ge 3).
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