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WorksheetYear 13Level 3 · Challenge
Free Year 13 The Binomial Distribution Worksheet
A senior maths worksheet on the binomial distribution: recognising Bernoulli trials and binomial situations, calculating P(X = k) with the binomial formula, cumulative and at least one probabilities, the mean np and variance np(1 − p), finding n and p from the mean and variance, and the smallest number of trials needed. Finishes with an extended seed germination problem. Full working is given for every answer.
Instant download · No sign-up · Free for personal, classroom and homeschool use
- Year
- Year 13
- Subject
- Maths
- Topic
- Statistics
- Difficulty
- Level 3 · Challenge
- Estimated time
- 50 minutes
- Curriculum
- New Zealand Curriculum
- Questions
- 16
- Answers
- Included, with working
- Format
- PDF (A4), print or complete online
Students will practise
- identifying the conditions for a binomial distribution and the parameters n and p
- calculating binomial probabilities with the binomial probability formula
- using the complement for at least and at most probabilities
- calculating and interpreting the mean, variance and standard deviation of a binomial random variable
Curriculum: New Zealand Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.
What's next?
Completed: The Binomial Distribution
How to use this worksheet
- Print the PDF or complete it on screen. Write the student's name and date at the top.
- Read the instructions for each part together before starting. Encourage working to be shown, not just answers.
- Mark with the answer sheet. Compare the method, not only the final answer, and give credit for correct working.
- Record the score, celebrate what went well, then follow the "What's next" suggestion.
Common questions
Who is this worksheet for?
Year 13 students (typically ages 17–18) working on statistics. It is pitched at level 3 · challenge.
Are the answers included?
Yes. The PDF includes a full answer sheet with the working for every question. You can also view the answers online.
How long does it take?
About 50 minutes for 16 questions. Short, regular sessions work best: two or three a week beats one long one.
Do I need to sign up to download?
No. Click Download Free PDF and it opens immediately. It is free for personal, classroom and homeschool use.
What should we do next?
Try Continuous Random Variables and Probability Density Functions.
Preview
Success TutoringWorksheetsYear 13 · Mathematics · StatisticsThe Binomial Distribution
Name
Date
Score /
Show your working. Give probabilities correct to 4 decimal places unless they are exact.
Part A: Bernoulli trials and binomial situations
A Bernoulli trial has exactly two outcomes, success (probability p) and failure (probability 1 − p). For one trial, E(X) = p and Var(X) = p(1 − p).
X is binomial, written X ~ Bin(n, p), when it counts the successes in a fixed number n of independent Bernoulli trials, each with the same probability p of success.
Worked example. A basketball player makes 70% of free throws and takes 10 shots, independently. The number of shots made is X ~ Bin(10, 0.7). The number of shots taken until the first miss is not binomial, because n is not fixed.
1.Which situation can be modelled by a binomial distribution?
- AThe number of sixes when a fair die is rolled 12 times
- BThe number of rolls of a die needed to get the first six
- CThe total of the numbers shown when a die is rolled 12 times
- DThe time until the next customer arrives
2.Drawing 5 cards without replacement from a standard 52-card deck and counting the hearts gives a binomial distribution. ☐ True ☐ False
3.A Bernoulli trial has probability of success p = 0.3. Find the mean and variance of the number of successes in one trial. (answer space)
Part B: Binomial probabilities
If X ~ Bin(n, p), then P(X = k) = [sup n]C[sup k] p[sup k] (1 − p)[sup n − k] for k = 0, 1, 2, …, n. The nCk counts the orders in which k successes can occur among the n trials.
For at least one, use the complement: P(X ≥ 1) = 1 − P(X = 0) = 1 − (1 − p)n. For at most k, add P(X = 0) up to P(X = k).
Worked example. A fair coin is tossed 4 times. P(exactly 3 heads) = 4C3 (0.5)3 (0.5)1 = 4 × = 0.25, and P(at least one head) = 1 − (0.5)4 = 1 − 0.0625 = 0.9375.
4.X ~ Bin(5, 0.5). Find P(X = 2).
5.X ~ Bin(10, 0.2). Find P(X = 3), correct to 4 decimal places.
6.X ~ Bin(6, 0.3). Find P(X = 0) and P(X ≥ 1), correct to 4 decimal places.
7.X ~ Bin(4, 0.6). Find P(X ≤ 1).
8.X ~ Bin(3, 0.4). Write the full probability distribution of X and check that the probabilities add to 1.
9.A multiple-choice quiz has 8 questions, each with 4 options. A student guesses every answer. Find, correct to 4 decimal places, (a) the probability of exactly 4 correct answers and (b) the probability of at least one correct answer.
10.A machine produces faulty items with probability 0.1, independently. What is the smallest number of items that must be checked for the probability of finding at least one faulty item to exceed 0.95?
Part C: Mean, variance and standard deviation
If X ~ Bin(n, p), then E(X) = np, Var(X) = np(1 − p) and SD(X) = √(np(1 − p)). The mean is the long-run average number of successes in n trials.
Worked example. A fair die is rolled 30 times and X is the number of sixes. E(X) = 30 × = 5, Var(X) = 30 × × = ≈ 4.17 and SD ≈ 2.04.
11.X ~ Bin(20, 0.15). Find the mean, the variance and the standard deviation of X (standard deviation to 2 decimal places).
12.A binomial random variable has mean 6 and variance 4.2. Find n and p.
Preview — the PDF contains the full resource and answer sheet.
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About this resource
- Created by
- Success Tutoring
- Last reviewed
- 1 October 2026
- How it was made
- Written by our team
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