Worksheets · Statistics

Independent and dependent events worksheet for Statistics

Statistics treats independence as something to check, not assume: A and B are independent exactly when P(A and B) equals P(A) × P(B). The bag questions set up the multiplication rule with and without replacement, and then the same test decides whether two events are related.

Also taught in Algebra 2, Geometry and Precalculus, 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

Independent and Dependent Events

Date Period

Find each probability, or answer as asked.

  1. Find the probability of independent events

    A spinner has 6 equal sections: 2 red and 4 blue. It is spun twice. Find the probability that it lands on blue both times.
  2. A game wheel has 11 equal sections: 5 green and 6 white. It is spun twice. Find the probability that it lands on green both times.
  3. A box holds 3 purple and 4 orange pens. One is drawn and put back before a second one is drawn. Find the probability that both are purple.
  4. A prize wheel has 8 equal sections: 3 gold and 5 silver. It is spun twice. Find the probability that it lands on gold both times.
  5. A cup holds 5 silver and 3 gold beads. Two are drawn with replacement. Find P(both silver).
  6. A game wheel has 6 equal sections: 4 green and 2 white. It is spun twice. Find the probability that it lands on white both times.
  7. A prize wheel has 5 equal sections: 2 gold and 3 silver. It is spun twice. Find the probability that it lands on silver both times.

  1. Find the probability without replacement

    A tin holds 13 brown and 17 white buttons. Two are drawn one at a time, and the first stays out. Find the probability that both are white.
  2. A bag has 11 red, 10 blue and 10 green counters. Two are drawn without replacement. Find the probability that neither is red.
  3. A class has 13 girls and 8 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.
  4. A rack holds 13 blue, 15 gray and 8 white shirts. One is removed and set aside, and then another is removed. Find P(both white).
  5. A team has 11 forwards and 8 defenders. The coach lets a computer pick a captain and then a co-captain at random. Find the probability that both are forwards.
  6. A basket has 25 tennis balls: 16 are yellow and the rest are pink. Two are drawn one after the other, and the first is not put back. Find the probability that both are pink.
  7. A tub holds 11 blue, 9 brown and 4 pink crayons. One is drawn and kept out, then a second one is drawn. Find the probability that neither is brown.

  1. Decide if the events are independent

    P(A)=0.1, P(B)=0.4, and P(A and B)=0.06. Are A and B independent?
  2. P(A)=0.2, P(B)=0.5, and P(A and B)=0.1. Are A and B independent?
  3. P(A)=0.9, P(B)=0.7, and P(A and B)=0.63. Are A and B independent?
  4. P(A)=0.7, P(B)=0.3, and P(A and B)=0.21. Are A and B independent?
  5. P(A)=0.6, P(B)=0.5, and P(A and B)=0.3. Are A and B independent?
  6. P(A)=0.4, P(B)=0.7, and P(A and B)=0.18. Are A and B independent?

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

Sigma Prep · sigmaprep.io/worksheets

Name

Independent and Dependent EventsAnswer keyVersion 3

Date Period

Find each probability, or answer as asked.

  1. Find the probability of independent events

    A spinner has 6 equal sections: 2 red and 4 blue. It is spun twice. Find the probability that it lands on blue both times.49
  2. A game wheel has 11 equal sections: 5 green and 6 white. It is spun twice. Find the probability that it lands on green both times.25121
  3. A box holds 3 purple and 4 orange pens. One is drawn and put back before a second one is drawn. Find the probability that both are purple.949
  4. A prize wheel has 8 equal sections: 3 gold and 5 silver. It is spun twice. Find the probability that it lands on gold both times.964
  5. A cup holds 5 silver and 3 gold beads. Two are drawn with replacement. Find P(both silver).2564
  6. A game wheel has 6 equal sections: 4 green and 2 white. It is spun twice. Find the probability that it lands on white both times.19
  7. A prize wheel has 5 equal sections: 2 gold and 3 silver. It is spun twice. Find the probability that it lands on silver both times.925

  1. Find the probability without replacement

    A tin holds 13 brown and 17 white buttons. Two are drawn one at a time, and the first stays out. Find the probability that both are white.136435
  2. A bag has 11 red, 10 blue and 10 green counters. Two are drawn without replacement. Find the probability that neither is red.3893
  3. A class has 13 girls and 8 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.1335
  4. A rack holds 13 blue, 15 gray and 8 white shirts. One is removed and set aside, and then another is removed. Find P(both white).245
  5. A team has 11 forwards and 8 defenders. The coach lets a computer pick a captain and then a co-captain at random. Find the probability that both are forwards.55171
  6. A basket has 25 tennis balls: 16 are yellow and the rest are pink. Two are drawn one after the other, and the first is not put back. Find the probability that both are pink.325
  7. A tub holds 11 blue, 9 brown and 4 pink crayons. One is drawn and kept out, then a second one is drawn. Find the probability that neither is brown.3592

  1. Decide if the events are independent

    P(A)=0.1, P(B)=0.4, and P(A and B)=0.06. Are A and B independent?No: P(A)P(B)=0.04≠0.06
  2. P(A)=0.2, P(B)=0.5, and P(A and B)=0.1. Are A and B independent?Yes: P(A)P(B)=0.1=P(A and B)
  3. P(A)=0.9, P(B)=0.7, and P(A and B)=0.63. Are A and B independent?Yes: P(A)P(B)=0.63=P(A and B)
  4. P(A)=0.7, P(B)=0.3, and P(A and B)=0.21. Are A and B independent?Yes: P(A)P(B)=0.21=P(A and B)
  5. P(A)=0.6, P(B)=0.5, and P(A and B)=0.3. Are A and B independent?Yes: P(A)P(B)=0.3=P(A and B)
  6. P(A)=0.4, P(B)=0.7, and P(A and B)=0.18. Are A and B independent?No: P(A)P(B)=0.28≠0.18

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

How to do these

P(A)=0.4, P(B)=0.5, and P(A and B)=0.25. Are A and B independent?

  1. If they were independent, P(A and B) would be 0.4 × 0.5 = 0.2.
  2. The actual value is 0.25, which is not 0.2.

The answer is No: P(A)P(B)=0.2≠0.25.

Where students go wrong

Reusing the first fraction after a draw without replacement. With 13 red among 21 counters, once a red is out there are 12 red among 20, so P(both red) is 13/21 × 12/20, not 13/21 × 13/21.

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

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
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) = 0.5 and P(A or B) = 0.85. Find P(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.

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 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.