Worksheets · Statistics

Probability of compound events worksheet for Statistics

Statistics writes these as formulas: P(A and B) = P(A) × P(B) for independent events, and P(A or B) = P(A) + P(B) - P(A and B) in general. The same rules, with the second draw adjusted when there is no replacement, lead into conditional probability.

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

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

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Name

Probability of Events

Date Period

Find each probability.

  1. Find the probability of independent events

    A bag has 3 red and 4 blue counters. After the first draw it goes back in, and a second one is drawn. Find the probability that both are blue.
  2. A cup holds 8 silver, 13 gold and 9 blue beads. One is taken, its color noted, and returned; then a second is taken. Find P(both gold).
  3. A spinner has 7 equal sections: 4 red and 3 blue. It is spun twice. Find the probability that it lands on blue both times.
  4. Box A holds 11 yellow and 17 pink tennis balls. Box B holds 11 yellow and 10 pink tennis balls. One is drawn from each box. Find the probability that both are yellow.

  1. Find the probability without replacement

    An office has 12 managers and 16 clerks. A lottery draws two of the workers, one at a time, to send to a training day. Find the probability that both are clerks.
  2. A pouch has 6 red and 5 green dice. One is drawn and kept out, then a second one is drawn. Find the probability that both are red.
  3. A bowl holds 10 red, 16 orange and 4 purple candies. Two are drawn one after the other, and the first is not put back. Find the probability that neither is red.
  4. A club has 2 seniors and 6 juniors. Two members are picked at random, one after the other, to plan the next meeting. Find the probability that both are seniors.

  1. Find P(A or B)P(A \text{ or } B)

    An 8-sided die is rolled. Find P(less than 3 or greater than 5).
  2. A 10-sided die is rolled. Find P(odd or greater than 6).
  3. An 8-sided die is rolled. Find P(odd or less than 6).
  4. A 10-sided die is rolled. Find P(even or greater than 8).

  1. Find the probability with combinations

    A box has 5 winning tickets out of 22. Three are drawn at once. Find P(all 3 are winners).
  2. A box has 6 winning tickets out of 10. Two are drawn at once. Find P(exactly one is a winner).
  3. A box has 8 winning tickets out of 15. Three are drawn at once. Find P(none of them win).
  4. A box has 3 winning tickets out of 11. Two are drawn at once. Find P(neither is a winner).

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

    A die is rolled 3 times. Find P(at least one 3).
  2. A coin is flipped 4 times. Find P(at least one head).
  3. A die is rolled 2 times. Find P(at least one 5).
  4. A spinner has 4 equal sections, 3 of them red. It is spun 3 times. Find P(at least one red).

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

Sigma Prep · sigmaprep.io/worksheets

Name

Probability of EventsAnswer keyVersion 4

Date Period

Find each probability.

  1. Find the probability of independent events

    A bag has 3 red and 4 blue counters. After the first draw it goes back in, and a second one is drawn. Find the probability that both are blue.1649
  2. A cup holds 8 silver, 13 gold and 9 blue beads. One is taken, its color noted, and returned; then a second is taken. Find P(both gold).169900
  3. A spinner has 7 equal sections: 4 red and 3 blue. It is spun twice. Find the probability that it lands on blue both times.949
  4. Box A holds 11 yellow and 17 pink tennis balls. Box B holds 11 yellow and 10 pink tennis balls. One is drawn from each box. Find the probability that both are yellow.121588

  1. Find the probability without replacement

    An office has 12 managers and 16 clerks. A lottery draws two of the workers, one at a time, to send to a training day. Find the probability that both are clerks.2063
  2. A pouch has 6 red and 5 green dice. One is drawn and kept out, then a second one is drawn. Find the probability that both are red.311
  3. A bowl holds 10 red, 16 orange and 4 purple candies. Two are drawn one after the other, and the first is not put back. Find the probability that neither is red.3887
  4. A club has 2 seniors and 6 juniors. Two members are picked at random, one after the other, to plan the next meeting. Find the probability that both are seniors.128

  1. Find P(A or B)P(A \text{ or } B)

    An 8-sided die is rolled. Find P(less than 3 or greater than 5).58
  2. A 10-sided die is rolled. Find P(odd or greater than 6).710
  3. An 8-sided die is rolled. Find P(odd or less than 6).34
  4. A 10-sided die is rolled. Find P(even or greater than 8).35

  1. Find the probability with combinations

    A box has 5 winning tickets out of 22. Three are drawn at once. Find P(all 3 are winners).1154
  2. A box has 6 winning tickets out of 10. Two are drawn at once. Find P(exactly one is a winner).815
  3. A box has 8 winning tickets out of 15. Three are drawn at once. Find P(none of them win).113
  4. A box has 3 winning tickets out of 11. Two are drawn at once. Find P(neither is a winner).2855

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

    A die is rolled 3 times. Find P(at least one 3).91216
  2. A coin is flipped 4 times. Find P(at least one head).1516
  3. A die is rolled 2 times. Find P(at least one 5).1136
  4. A spinner has 4 equal sections, 3 of them red. It is spun 3 times. Find P(at least one red).6364

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

How to do these

A 12-sided die is rolled. Find P(even or a multiple of 3).

  1. P(even) = 6/12 and P(multiple of 3) = 4/12.
  2. Both happen for 6 and 12, so P(both) = 2/12.
  3. 6/12 + 4/12 - 2/12 = 8/12.

The answer is 23.

Where students go wrong

Adding without taking off the overlap. On a 12-sided die, 6 and 12 are both even and multiples of 3, and counting them twice gives 10/12 instead of 8/12.

Also called independent events, dependent events or compound probability.

What you can put on this worksheet

Independent events: both the same
A spinner has 9 equal sections: 4 red and 5 blue. It is spun twice. Find the probability that it lands on red both times. → 1681
Independent events: one of each
A spinner has 9 equal sections: 4 red and 5 blue. It is spun twice. Find the probability that it lands on red once and blue once. → 4081
Dependent events: both the same
A class has 4 girls and 5 boys. To pick who hands out papers, the teacher draws two names from a hat, one after the other. Find the probability that both are girls. → 16
Dependent events: one of each
A class has 4 girls and 5 boys. To pick who hands out papers, the teacher draws two names from a hat, one after the other. Find the probability that the two are one of the girls and one of the boys, in either order. → 59
One event or the other
A 6-sided die is rolled. Find P(greater than 3 or less than 3). → 56
Probability with combinations
A box has 3 winning tickets out of 10. Two are drawn at once. Find P(neither is a winner). → 715
Experimental probability
A spinner was spun 40 times and landed on red on 22 of the spins. Find the experimental probability of landing on red. → 1120
At least one, by the complement
A die is rolled 3 times. Find P(at least one 5). → 91216
Two different experiments: a coin and a die, a die twice
A spinner with 5 equal sections numbered 1 to 5 is spun twice. Find P(a 5, then an even number). → 225
Three picks in a set order
A drawer has 4 black, 2 gray and 3 white socks. Three are taken at random, one after another, without putting any back. Find P(black, then black, then black). → 121
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
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

Questions about these worksheets

Yes. Every probability of compound events 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 of compound events 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: independent events: both the same, independent events: one of each, dependent events: both the same, dependent events: one of each, one event or the other, probability with combinations, experimental probability, at least one, by the complement, two different experiments: a coin and a die, a die twice, three picks in a set order, drawing from two groups: exactly so many and coins, dice and guessing: exactly, at least, at most. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the same fraction twice when there is no replacement. The second draw happens in a smaller bag with one fewer of whatever you took, and forgetting that is the single most common error on this topic.