Worksheets · Statistics

Probability with combinatorics worksheet for Statistics

Statistics uses these counting methods to build two models: drawing without replacement, counted with combinations from each group, and the binomial setting, with n independent trials and a fixed chance of success. The complement handles 'at least one' questions much faster than adding up cases.

Also taught in Precalculus and Algebra 2, each at its own level and with its own examples.

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Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Probability with Combinatorics

Date Period

Answer each question. Leave probabilities as fractions.

  1. Find P(exactly k)P(\text{exactly } k) in a random draw

    A batch has 2 defective parts and 4 good parts. Three parts are inspected at random. Find P(exactly 2 defective).
  2. A standard 52-card deck has 26 red cards and 26 black cards. Three cards are dealt at random. Find P(exactly 3 red).
  3. A class has 6 girls and 6 boys. Two students are chosen at random. Find P(exactly 1 girl).
  4. A club has 4 seniors and 4 juniors. Three members are picked at random. Find P(exactly 3 seniors).

  1. Find P(at least one)P(\text{at least one}) in a random draw

    A batch has 3 defective parts and 13 good parts. Four parts are inspected at random. Find P(at least one defective).
  2. A class has 12 girls and 13 boys. Two students are chosen at random. Find P(at least one girl).
  3. A bag holds 12 red marbles and 18 blue marbles. Three marbles are drawn at random. Find P(at least one red).
  4. A standard 52-card deck has 26 red cards and 26 black cards. Four cards are dealt at random. Find P(at least one red).

  1. Find P(exactly r successes)P(\text{exactly } r \text{ successes})

    An archer shoots 4 arrows and hits the target with each one, independently, with probability 14. Find P(exactly 3 hits).
  2. A player plays 5 games and wins each one, independently, with probability 12. Find P(exactly 4 wins).
  3. A player plays 5 games and wins each one, independently, with probability 34. Find P(exactly 1 win).

  1. Find P(at least one success)P(\text{at least one success})

    A player plays 6 games and wins each one, independently, with probability 23. Find P(at least one win).
  2. A basketball player takes 5 free throws and makes each one, independently, with probability 16. Find P(at least one made shot).
  3. An archer shoots 5 arrows and hits the target with each one, independently, with probability 15. Find P(at least one hit).

  1. Find the binomial mean and standard deviation

    X counts successes in 38 independent trials, each with probability 0.2 of success. Find the mean of X and its standard deviation, to the nearest hundredth.
  2. Each shopper who walks into a store buys something with probability 0.3, independently. Of the next 40 shoppers, X is the number who buy. What are the mean and the standard deviation of X, to the nearest hundredth?
  3. A player makes each free throw with probability 0.4, independently. She shoots 28 free throws, and X is the number she makes. Find the mean of X and its standard deviation, to the nearest hundredth.

  1. Find probabilities for random line-ups

    A group of 4 people, including Ana and Ben, line up in a random order. What is the probability that Ana is first in line?
  2. A group of 8 people, including Ana and Ben, line up in a random order. What is the probability that Ana and Ben stand at the two ends of the line?
  3. A group of 5 people, including Ana and Ben, line up in a random order. What is the probability that Ana and Ben do not stand next to each other?

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Probability with CombinatoricsAnswer keyVersion 2

Date Period

Answer each question. Leave probabilities as fractions.

  1. Find P(exactly k)P(\text{exactly } k) in a random draw

    A batch has 2 defective parts and 4 good parts. Three parts are inspected at random. Find P(exactly 2 defective).15
  2. A standard 52-card deck has 26 red cards and 26 black cards. Three cards are dealt at random. Find P(exactly 3 red).217
  3. A class has 6 girls and 6 boys. Two students are chosen at random. Find P(exactly 1 girl).611
  4. A club has 4 seniors and 4 juniors. Three members are picked at random. Find P(exactly 3 seniors).114

  1. Find P(at least one)P(\text{at least one}) in a random draw

    A batch has 3 defective parts and 13 good parts. Four parts are inspected at random. Find P(at least one defective).1728
  2. A class has 12 girls and 13 boys. Two students are chosen at random. Find P(at least one girl).3750
  3. A bag holds 12 red marbles and 18 blue marbles. Three marbles are drawn at random. Find P(at least one red).8111015
  4. A standard 52-card deck has 26 red cards and 26 black cards. Four cards are dealt at random. Find P(at least one red).787833

  1. Find P(exactly r successes)P(\text{exactly } r \text{ successes})

    An archer shoots 4 arrows and hits the target with each one, independently, with probability 14. Find P(exactly 3 hits).364
  2. A player plays 5 games and wins each one, independently, with probability 12. Find P(exactly 4 wins).532
  3. A player plays 5 games and wins each one, independently, with probability 34. Find P(exactly 1 win).151024

  1. Find P(at least one success)P(\text{at least one success})

    A player plays 6 games and wins each one, independently, with probability 23. Find P(at least one win).728729
  2. A basketball player takes 5 free throws and makes each one, independently, with probability 16. Find P(at least one made shot).46517776
  3. An archer shoots 5 arrows and hits the target with each one, independently, with probability 15. Find P(at least one hit).21013125

  1. Find the binomial mean and standard deviation

    X counts successes in 38 independent trials, each with probability 0.2 of success. Find the mean of X and its standard deviation, to the nearest hundredth.mean 7.6, SD 2.47
  2. Each shopper who walks into a store buys something with probability 0.3, independently. Of the next 40 shoppers, X is the number who buy. What are the mean and the standard deviation of X, to the nearest hundredth?mean 12, SD 2.90
  3. A player makes each free throw with probability 0.4, independently. She shoots 28 free throws, and X is the number she makes. Find the mean of X and its standard deviation, to the nearest hundredth.mean 11.2, SD 2.59

  1. Find probabilities for random line-ups

    A group of 4 people, including Ana and Ben, line up in a random order. What is the probability that Ana is first in line?14
  2. A group of 8 people, including Ana and Ben, line up in a random order. What is the probability that Ana and Ben stand at the two ends of the line?128
  3. A group of 5 people, including Ana and Ben, line up in a random order. What is the probability that Ana and Ben do not stand next to each other?35

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

An archer shoots 4 arrows and hits the target with each one, independently, with probability 13. Find P(exactly 2 hits).

  1. There are C(4, 2) = 6 ways to place the 2 hits.
  2. Each way has probability (1/3)² × (2/3)² = 4/81.
  3. So the probability is 6 × 4/81 = 24/81.

The answer is 827.

Where students go wrong

Forgetting the orders the successes can fall in. Exactly 2 hits in 4 shots is not one outcome: C(4, 2) = 6 orders each have the same chance, so the single-order probability is multiplied by 6.

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 standard 52-card deck has 26 red cards and 26 black cards. Three cards are dealt at random. Find P(exactly 3 red). → 217
Drawing from two groups: at least one
A standard 52-card deck has 26 red cards and 26 black cards. Three cards are dealt at random. Find P(at least one red). → 1517
Exactly r successes in n trials
An archer shoots 5 arrows and hits the target with each one, independently, with probability 12. Find P(exactly 1 hit). → 532
At least one success in n trials
An archer shoots 5 arrows and hits the target with each one, independently, with probability 12. Find P(at least one hit). → 3132
At least r successes
An archer shoots 4 arrows and hits the target with each one, independently, with probability 12. Find P(at least one hit). → 1516
At most r successes
An archer shoots 4 arrows and hits the target with each one, independently, with probability 12. Find P(at most one hit). → 516
Mean and SD of a binomial count
Each shopper who walks into a store buys something with probability 0.4, independently. Of the next 26 shoppers, X is the number who buy. Find the mean of X and its standard deviation, to the nearest hundredth. → mean 10.4, SD 2.50
Arrangements in a line
A group of 6 people, including Ana and Ben, line up in a random order. What is the probability that Ana and Ben stand at the two ends of the line? → 115
Drawing from two groups: at most so many
A class has 4 girls and 7 boys. Three students are picked at random. Find P(at most 2 girls). → 161165
Coins, dice and guessing: exactly, at least, at most
A student guesses on every question of a 4-question true-false quiz. Find P(at most 1 right answer). → 516
None of a group chosen
A committee of 7 is picked at random from 15 students. Of the students, 4 are seniors. What is the probability that at least one senior is on the committee? → 3739
p from a percentage or a ratio
Of all customers, 90% pay with a card. 6 customers are picked at random. Find P(exactly 3 pay with a card), to four decimal places. → 0.0146

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.

Statistics 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 12: 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, at most r successes, mean and sd of a binomial count, arrangements in a line, drawing from two groups: at most so many, coins, dice and guessing: exactly, at least, at most, none of a group chosen and p from a percentage or a ratio. Tick as many as you want and set how many of each, or let it spread them evenly.

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.