Worksheets · Statistics

Geometric distribution worksheet

A geometric count is the number of trials until the first success. The first success on trial k means k - 1 failures and then a success, and the expected wait is 1/p trials.

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

Sigma Prep · sigmaprep.io/worksheets

Name

The Geometric Distribution

Date Period

Answer each question.

  1. Each trial succeeds with probability 0.4, independently. Find the probability that the first success is on trial 4, to four decimal places.
  2. Each trial succeeds with probability 0.5, independently. What is the expected number of trials until the first success?
  3. Each trial succeeds with probability 0.4, independently. What is the expected number of trials until the first success?
  4. Each trial succeeds with probability 0.5, independently. Find the probability that the first success comes within the first 4 trials, to four decimal places.
  5. Each trial succeeds with probability 0.2, independently. Find the probability that the first success comes within the first 3 trials, to four decimal places.
  6. Each trial succeeds with probability 0.2, independently. Find the probability that the first success comes within the first 2 trials, to four decimal places.
  7. Each trial succeeds with probability 0.25, independently. What is the expected number of trials until the first success?
  8. Each trial succeeds with probability 0.1, independently. Find the probability that the first success is on trial 4, to four decimal places.
  9. Each trial succeeds with probability 0.1, independently. Find the probability that the first success comes within the first 4 trials, to four decimal places.
  10. Each trial succeeds with probability 0.3, independently. Find the probability that the first success is on trial 5, to four decimal places.
  11. Each trial succeeds with probability 0.1, independently. What is the expected number of trials until the first success?
  12. Each trial succeeds with probability 0.1, independently. Find the probability that the first success comes within the first 3 trials, to four decimal places.
  13. Each trial succeeds with probability 0.2, independently. What is the expected number of trials until the first success?
  14. Each trial succeeds with probability 0.2, independently. Find the probability that the first success is on trial 2, to four decimal places.
  15. Each trial succeeds with probability 0.5, independently. Find the probability that the first success comes within the first 6 trials, to four decimal places.
  16. Each trial succeeds with probability 0.5, independently. Find the probability that the first success is on trial 6, to four decimal places.
  17. Each trial succeeds with probability 0.25, independently. Find the probability that the first success is on trial 5, to four decimal places.
  18. Each trial succeeds with probability 0.4, independently. Find the probability that the first success comes within the first 2 trials, to four decimal places.
  19. Each trial succeeds with probability 0.3, independently. What is the expected number of trials until the first success?
  20. Each trial succeeds with probability 0.2, independently. Find the probability that the first success is on trial 6, to four decimal places.

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

Sigma Prep · sigmaprep.io/worksheets

Name

The Geometric DistributionAnswer keyVersion 1

Date Period

Answer each question.

  1. Each trial succeeds with probability 0.4, independently. Find the probability that the first success is on trial 4, to four decimal places.0.0864
  2. Each trial succeeds with probability 0.5, independently. What is the expected number of trials until the first success?2
  3. Each trial succeeds with probability 0.4, independently. What is the expected number of trials until the first success?52
  4. Each trial succeeds with probability 0.5, independently. Find the probability that the first success comes within the first 4 trials, to four decimal places.0.9375
  5. Each trial succeeds with probability 0.2, independently. Find the probability that the first success comes within the first 3 trials, to four decimal places.0.4880
  6. Each trial succeeds with probability 0.2, independently. Find the probability that the first success comes within the first 2 trials, to four decimal places.0.3600
  7. Each trial succeeds with probability 0.25, independently. What is the expected number of trials until the first success?4
  8. Each trial succeeds with probability 0.1, independently. Find the probability that the first success is on trial 4, to four decimal places.0.0729
  9. Each trial succeeds with probability 0.1, independently. Find the probability that the first success comes within the first 4 trials, to four decimal places.0.3439
  10. Each trial succeeds with probability 0.3, independently. Find the probability that the first success is on trial 5, to four decimal places.0.0720
  11. Each trial succeeds with probability 0.1, independently. What is the expected number of trials until the first success?10
  12. Each trial succeeds with probability 0.1, independently. Find the probability that the first success comes within the first 3 trials, to four decimal places.0.2710
  13. Each trial succeeds with probability 0.2, independently. What is the expected number of trials until the first success?5
  14. Each trial succeeds with probability 0.2, independently. Find the probability that the first success is on trial 2, to four decimal places.0.1600
  15. Each trial succeeds with probability 0.5, independently. Find the probability that the first success comes within the first 6 trials, to four decimal places.0.9844
  16. Each trial succeeds with probability 0.5, independently. Find the probability that the first success is on trial 6, to four decimal places.0.0156
  17. Each trial succeeds with probability 0.25, independently. Find the probability that the first success is on trial 5, to four decimal places.0.0791
  18. Each trial succeeds with probability 0.4, independently. Find the probability that the first success comes within the first 2 trials, to four decimal places.0.6400
  19. Each trial succeeds with probability 0.3, independently. What is the expected number of trials until the first success?103
  20. Each trial succeeds with probability 0.2, independently. Find the probability that the first success is on trial 6, to four decimal places.0.0655

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

How to do these

Each trial succeeds with probability 0.2, independently. Find the probability that the first success is on trial 4, to four decimal places.

  1. The first three trials fail, each with probability 0.8.
  2. The fourth succeeds, with probability 0.2.
  3. 0.8³ times 0.2.

The answer is 0.1024.

Where students go wrong

Using k failures instead of k - 1. The success is the kth trial itself.

Also called geometric probability, waiting time distribution or first success.

Questions about these worksheets

Yes. Every geometric distribution 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 geometric distribution 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.

Using k failures instead of k - 1. The success is the kth trial itself.