Worksheets · Precalculus

Precalculus independent and dependent events worksheet

Precalculus puts probability next to counting, so many of these are solved with combinations: favorable outcomes over total outcomes. Drawing without replacement is the same as choosing a group all at once, and the multiplication rule should give the same answer.

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

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Name

Independent and Dependent Events

Date Period

Find each probability, or answer as asked.

  1. A bag has 14 red and 8 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).
  2. A bag has 13 red and 18 blue counters. Two are drawn without replacement. Find P(both red).
  3. P(A)=0.4, P(B)=0.9, and P(A and B)=0.36. Are A and B independent?
  4. A bag has 13 red and 9 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).
  5. A bag has 9 red and 15 blue counters. Two are drawn without replacement. Find P(both red).
  6. P(A)=0.7, P(B)=0.3, and P(A and B)=0.26. Are A and B independent?
  7. A bag has 12 red and 18 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).
  8. A bag has 8 red and 11 blue counters. Two are drawn without replacement. Find P(both red).
  9. P(A)=0.4, P(B)=0.3, and P(A and B)=0.14. Are A and B independent?
  10. A bag has 13 red and 10 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).
  11. A bag has 11 red and 15 blue counters. Two are drawn without replacement. Find P(both red).
  12. P(A)=0.9, P(B)=0.6, and P(A and B)=0.59. Are A and B independent?
  13. A bag has 8 red and 17 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).
  14. A bag has 10 red and 15 blue counters. Two are drawn without replacement. Find P(both red).
  15. P(A)=0.6, P(B)=0.8, and P(A and B)=0.58. Are A and B independent?
  16. A bag has 15 red and 11 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).
  17. A bag has 10 red and 13 blue counters. Two are drawn without replacement. Find P(both red).
  18. P(A)=0.1, P(B)=0.6, and P(A and B)=0.08. Are A and B independent?
  19. A bag has 14 red and 13 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).
  20. A bag has 13 red and 17 blue counters. Two are drawn without replacement. Find P(both red).

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

Sigma Prep · sigmaprep.io/worksheets

Name

Independent and Dependent EventsAnswer keyVersion 1

Date Period

Find each probability, or answer as asked.

  1. A bag has 14 red and 8 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).49121
  2. A bag has 13 red and 18 blue counters. Two are drawn without replacement. Find P(both red).26155
  3. P(A)=0.4, P(B)=0.9, and P(A and B)=0.36. Are A and B independent?Yes: P(A)P(B)=0.36=P(A and B)
  4. A bag has 13 red and 9 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).169484
  5. A bag has 9 red and 15 blue counters. Two are drawn without replacement. Find P(both red).323
  6. P(A)=0.7, P(B)=0.3, and P(A and B)=0.26. Are A and B independent?No: P(A)P(B)=0.21≠0.26
  7. A bag has 12 red and 18 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).425
  8. A bag has 8 red and 11 blue counters. Two are drawn without replacement. Find P(both red).28171
  9. P(A)=0.4, P(B)=0.3, and P(A and B)=0.14. Are A and B independent?No: P(A)P(B)=0.12≠0.14
  10. A bag has 13 red and 10 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).169529
  11. A bag has 11 red and 15 blue counters. Two are drawn without replacement. Find P(both red).1165
  12. P(A)=0.9, P(B)=0.6, and P(A and B)=0.59. Are A and B independent?No: P(A)P(B)=0.54≠0.59
  13. A bag has 8 red and 17 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).64625
  14. A bag has 10 red and 15 blue counters. Two are drawn without replacement. Find P(both red).320
  15. P(A)=0.6, P(B)=0.8, and P(A and B)=0.58. Are A and B independent?No: P(A)P(B)=0.48≠0.58
  16. A bag has 15 red and 11 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).225676
  17. A bag has 10 red and 13 blue counters. Two are drawn without replacement. Find P(both red).45253
  18. P(A)=0.1, P(B)=0.6, and P(A and B)=0.08. Are A and B independent?No: P(A)P(B)=0.06≠0.08
  19. A bag has 14 red and 13 blue counters. One is drawn, replaced, then another is drawn. Find P(both red).196729
  20. A bag has 13 red and 17 blue counters. Two are drawn without replacement. Find P(both red).26145

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

How to do these

Two cards are drawn from a standard deck without replacement. Find P(both aces)

  1. There are C(52, 2) = 1326 possible pairs.
  2. C(4, 2) = 6 of them are two aces, so the chance is 6/1326.
  3. Check with the multiplication rule: 4/52 × 3/51 = 12/2652, the same.

The answer is 1221.

Where students go wrong

Counting the top and bottom in different ways, such as ordered pairs on top and combinations on the bottom. Use combinations for both or permutations for both, and the answer matches the multiplication rule.

Also called with and without replacement, compound probability or multiplying probabilities.

What you can put on this worksheet

Independent events: both the same
A bag has 4 red and 4 blue counters. One is drawn, replaced, then another is drawn. Find P(both red). → 14
Independent events: one of each
A bag has 4 red and 4 blue counters. One is drawn, replaced, then another is drawn. Find P(one of each color). → 12
Dependent events: both the same
A bag has 4 red and 4 blue counters. Two are drawn without replacement. Find P(both red). → 314
Dependent events: one of each
A bag has 4 red and 4 blue counters. Two are drawn without replacement. Find P(one of each color). → 47
Independent or dependent?
A spinner is spun twice. A: the first spin lands on blue. B: the second spin lands on blue. Are A and B independent or dependent? → Independent
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
Independent? From P(A | B) or P(not A)
P(A)=0.5, P(B)=0.6, and P(A∣B)=0.6. Are A and B independent? → No: P(A∣B)≠P(A)
Independent events: find the missing probability
A and B are independent. P(A)=0.5 and P(not B)=0.4. Find P(A and B) → 0.3
Conditional probability with P(A or B)
P(A)=0.6, P(B)=0.6, and P(A or B)=0.96. Find P(A∣B) → 0.4
Working back to a probability
A and B are independent. P(A) = 0.6 and P(A and B) = 0.18. Find P(not B). → 0.7

Questions about these worksheets

Yes. Every independent and dependent 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.

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 independent and dependent 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 three 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 11: independent events: both the same, independent events: one of each, dependent events: both the same, dependent events: one of each, independent or dependent?, two different experiments: a coin and a die, a die twice, three picks in a set order, independent? from p(a | b) or p(not a), independent events: find the missing probability, conditional probability with p(a or b) and working back to a probability. Tick as many as you want and set how many of each, or let it spread them evenly.

Keeping the same denominator on the second draw. Without replacement there is one fewer of everything, and using the original total is what makes a dependent question come out as an independent one.