Probability with combinatorics worksheet

Counting the ways something can happen and dividing by the ways anything can happen is all a probability is. With repeated trials the count is a binomial coefficient, which is why this sits next to the counting work.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Probability with Combinatorics Date Period Find each probability. Leave answers as fractions. A batch has 9 defective parts and 14 good parts. 4 are inspected. $P ($ exactly 4 defective $)$ A class has 15 girls and 16 boys. 3 are chosen. $P ($ at least one girl $)$ 6 trials, $P ($ success $) = \frac{2}{5}$ . $P ($ exactly 2 successes $)$ 7 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ at least one success $)$ A class has 16 girls and 11 boys. 2 are chosen. $P ($ exactly 1 girl $)$ A deck has 14 red cards and 16 black cards. 2 are dealt. $P ($ at least one red $)$ 9 trials, $P ($ success $) = \frac{3}{5}$ . $P ($ exactly 1 success $)$ 9 trials, $P ($ success $) = \frac{1}{2}$ . $P ($ at least one success $)$ A class has 16 girls and 10 boys. 2 are chosen. $P ($ exactly 1 girl $)$ A class has 10 girls and 16 boys. 3 are chosen. $P ($ at least one girl $)$ 8 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ exactly 7 successes $)$ 8 trials, $P ($ success $) = \frac{2}{3}$ . $P ($ at least one success $)$ A class has 15 girls and 18 boys. 4 are chosen. $P ($ exactly 3 girls $)$ A club has 17 seniors and 18 juniors. 2 are picked. $P ($ at least one senior $)$ 9 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ exactly 5 successes $)$ 6 trials, $P ($ success $) = \frac{2}{3}$ . $P ($ at least one success $)$ A club has 10 seniors and 18 juniors. 2 are picked. $P ($ exactly 1 senior $)$ A bag holds 13 red marbles and 12 blue marbles. 3 are drawn. $P ($ at least one red $)$ 9 trials, $P ($ success $) = \frac{1}{5}$ . $P ($ exactly 8 successes $)$ 8 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ at least one success $)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Probability with Combinatorics

Date Period

Find each probability. Leave answers as fractions.

A batch has 9 defective parts and 14 good parts. 4 are inspected. $P ($ exactly 4 defective $)$
A class has 15 girls and 16 boys. 3 are chosen. $P ($ at least one girl $)$
6 trials, $P ($ success $) = \frac{2}{5}$ . $P ($ exactly 2 successes $)$
7 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ at least one success $)$
A class has 16 girls and 11 boys. 2 are chosen. $P ($ exactly 1 girl $)$
A deck has 14 red cards and 16 black cards. 2 are dealt. $P ($ at least one red $)$
9 trials, $P ($ success $) = \frac{3}{5}$ . $P ($ exactly 1 success $)$
9 trials, $P ($ success $) = \frac{1}{2}$ . $P ($ at least one success $)$
A class has 16 girls and 10 boys. 2 are chosen. $P ($ exactly 1 girl $)$
A class has 10 girls and 16 boys. 3 are chosen. $P ($ at least one girl $)$
8 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ exactly 7 successes $)$
8 trials, $P ($ success $) = \frac{2}{3}$ . $P ($ at least one success $)$
A class has 15 girls and 18 boys. 4 are chosen. $P ($ exactly 3 girls $)$
A club has 17 seniors and 18 juniors. 2 are picked. $P ($ at least one senior $)$
9 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ exactly 5 successes $)$
6 trials, $P ($ success $) = \frac{2}{3}$ . $P ($ at least one success $)$
A club has 10 seniors and 18 juniors. 2 are picked. $P ($ exactly 1 senior $)$
A bag holds 13 red marbles and 12 blue marbles. 3 are drawn. $P ($ at least one red $)$
9 trials, $P ($ success $) = \frac{1}{5}$ . $P ($ exactly 8 successes $)$
8 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ at least one success $)$



Sigma Prep · sigmaprep.io/worksheets Name Probability with CombinatoricsAnswer keyVersion 1 Date Period Find each probability. Leave answers as fractions. A batch has 9 defective parts and 14 good parts. 4 are inspected. $P ($ exactly 4 defective $)$ $\frac{18}{1265}$ A class has 15 girls and 16 boys. 3 are chosen. $P ($ at least one girl $)$ $\frac{787}{899}$ 6 trials, $P ($ success $) = \frac{2}{5}$ . $P ($ exactly 2 successes $)$ $\frac{972}{3125}$ 7 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ at least one success $)$ $\frac{2059}{2187}$ A class has 16 girls and 11 boys. 2 are chosen. $P ($ exactly 1 girl $)$ $\frac{176}{351}$ A deck has 14 red cards and 16 black cards. 2 are dealt. $P ($ at least one red $)$ $\frac{21}{29}$ 9 trials, $P ($ success $) = \frac{3}{5}$ . $P ($ exactly 1 success $)$ $\frac{6912}{1953125}$ 9 trials, $P ($ success $) = \frac{1}{2}$ . $P ($ at least one success $)$ $\frac{511}{512}$ A class has 16 girls and 10 boys. 2 are chosen. $P ($ exactly 1 girl $)$ $\frac{32}{65}$ A class has 10 girls and 16 boys. 3 are chosen. $P ($ at least one girl $)$ $\frac{51}{65}$ 8 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ exactly 7 successes $)$ $\frac{16}{6561}$ 8 trials, $P ($ success $) = \frac{2}{3}$ . $P ($ at least one success $)$ $\frac{6560}{6561}$ A class has 15 girls and 18 boys. 4 are chosen. $P ($ exactly 3 girls $)$ $\frac{273}{1364}$ A club has 17 seniors and 18 juniors. 2 are picked. $P ($ at least one senior $)$ $\frac{26}{35}$ 9 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ exactly 5 successes $)$ $\frac{224}{2187}$ 6 trials, $P ($ success $) = \frac{2}{3}$ . $P ($ at least one success $)$ $\frac{728}{729}$ A club has 10 seniors and 18 juniors. 2 are picked. $P ($ exactly 1 senior $)$ $\frac{10}{21}$ A bag holds 13 red marbles and 12 blue marbles. 3 are drawn. $P ($ at least one red $)$ $\frac{104}{115}$ 9 trials, $P ($ success $) = \frac{1}{5}$ . $P ($ exactly 8 successes $)$ $\frac{36}{1953125}$ 8 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ at least one success $)$ $\frac{6305}{6561}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Probability with CombinatoricsAnswer keyVersion 1

Date Period

Find each probability. Leave answers as fractions.

A batch has 9 defective parts and 14 good parts. 4 are inspected. $P ($ exactly 4 defective $)$ $\frac{18}{1265}$
A class has 15 girls and 16 boys. 3 are chosen. $P ($ at least one girl $)$ $\frac{787}{899}$
6 trials, $P ($ success $) = \frac{2}{5}$ . $P ($ exactly 2 successes $)$ $\frac{972}{3125}$
7 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ at least one success $)$ $\frac{2059}{2187}$
A class has 16 girls and 11 boys. 2 are chosen. $P ($ exactly 1 girl $)$ $\frac{176}{351}$
A deck has 14 red cards and 16 black cards. 2 are dealt. $P ($ at least one red $)$ $\frac{21}{29}$
9 trials, $P ($ success $) = \frac{3}{5}$ . $P ($ exactly 1 success $)$ $\frac{6912}{1953125}$
9 trials, $P ($ success $) = \frac{1}{2}$ . $P ($ at least one success $)$ $\frac{511}{512}$
A class has 16 girls and 10 boys. 2 are chosen. $P ($ exactly 1 girl $)$ $\frac{32}{65}$
A class has 10 girls and 16 boys. 3 are chosen. $P ($ at least one girl $)$ $\frac{51}{65}$
8 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ exactly 7 successes $)$ $\frac{16}{6561}$
8 trials, $P ($ success $) = \frac{2}{3}$ . $P ($ at least one success $)$ $\frac{6560}{6561}$
A class has 15 girls and 18 boys. 4 are chosen. $P ($ exactly 3 girls $)$ $\frac{273}{1364}$
A club has 17 seniors and 18 juniors. 2 are picked. $P ($ at least one senior $)$ $\frac{26}{35}$
9 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ exactly 5 successes $)$ $\frac{224}{2187}$
6 trials, $P ($ success $) = \frac{2}{3}$ . $P ($ at least one success $)$ $\frac{728}{729}$
A club has 10 seniors and 18 juniors. 2 are picked. $P ($ exactly 1 senior $)$ $\frac{10}{21}$
A bag holds 13 red marbles and 12 blue marbles. 3 are drawn. $P ($ at least one red $)$ $\frac{104}{115}$
9 trials, $P ($ success $) = \frac{1}{5}$ . $P ($ exactly 8 successes $)$ $\frac{36}{1953125}$
8 trials, $P ($ success $) = \frac{1}{3}$ . $P ($ at least one success $)$ $\frac{6305}{6561}$

How to do these

A bag has 5 red and 7 blue. 3 are drawn. $P ($ exactly 2 red $)$

Choose 2 of the 5 red: 10 ways.
Choose 1 of the 7 blue: 7 ways.
Out of 220 ways to choose any 3.

The answer is $\frac{7}{22}$ .

Where students go wrong

Adding probabilities that overlap, or forgetting the ways the successes can be arranged. Exactly 2 successes in 5 trials is not one outcome: there are 10 orders it can happen in, and the binomial coefficient is what counts them.

Also called binomial probability, combinations and probability or at least probability.

What you can put on this worksheet

Drawing from two groups: exactly so many: A deck has 5 red cards and 5 black cards. 2 are dealt. $P ($ exactly 1 red $)$ → $\frac{5}{9}$
Drawing from two groups: at least one: A deck has 5 red cards and 5 black cards. 2 are dealt. $P ($ at least one red $)$ → $\frac{7}{9}$
Exactly r successes in n trials: 5 trials, $P ($ success $) = \frac{1}{2}$ . $P ($ exactly 0 successes $)$ → $\frac{1}{32}$
At least one success in n trials: 5 trials, $P ($ success $) = \frac{1}{2}$ . $P ($ at least one success $)$ → $\frac{31}{32}$
At least r successes: 4 trials, $P ($ success $) = \frac{1}{2}$ . $P ($ at least 1 success $)$ → $\frac{15}{16}$
At most r successes: 4 trials, $P ($ success $) = \frac{1}{2}$ . $P ($ at most 1 success $)$ → $\frac{5}{16}$

Questions about these worksheets

Yes. Every probability with combinatorics sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Precalculus is usually taken around 11th or 12th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the probability with combinatorics questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: drawing from two groups: exactly so many, drawing from two groups: at least one, exactly r successes in n trials, at least one success in n trials, at least r successes and at most r successes. Tick as many as you want and set how many of each, or let it spread them evenly.

Probability with combinatorics worksheet

Select your difficulty

Probability with Combinatorics

Probability with CombinatoricsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these probability with combinatorics worksheets free?+

Do the probability with combinatorics worksheets come with an answer key?+

Are these probability with combinatorics worksheets printable?+

What grade are these probability with combinatorics worksheets for?+

Can I get a different version of the same probability with combinatorics worksheet?+

Is this a free alternative to Kuta Software for probability with combinatorics?+

Can I choose how hard the probability with combinatorics questions are?+

What kinds of question are on a probability with combinatorics worksheet?+

What do students get wrong with probability with combinatorics?+