Study Guide
Probability: Tree Diagrams and Conditional Probability
An HSC/VCE-style study guide to multi-stage probability: the addition and multiplication rules, tree diagrams with and without…
- Difficulty
- Level 2 · Standard
- Time
- 25 min
WorksheetYear 11Level 3 · ChallengeHSCVCE
Free Year 11 Discrete Random Variables and Expected Value Worksheet
A senior maths worksheet on discrete probability distributions: checking that a table is a valid distribution, finding unknown probabilities, calculating the expected value, variance and standard deviation, deciding whether a game is fair and building a distribution from coins and counters. Finishes with an extended problem on the daily profit of a market stall. Full working is given for every answer.
Instant download · No sign-up · Free for personal, classroom and homeschool use
- Year
- Year 11
- Subject
- Maths
- Topic
- Probability
- Difficulty
- Level 3 · Challenge
- Estimated time
- 45 minutes
- Curriculum
- Australian Curriculum
- Questions
- 15
- Answers
- Included, with working
- Format
- PDF (A4), print or complete online
Students will practise
- recognising a valid discrete probability distribution and finding a missing probability
- calculating the expected value E(X) = Σ x P(X = x) and interpreting it as a long-run average
- calculating the variance and standard deviation of a discrete random variable
- using expected value to judge whether a game or raffle is fair
Curriculum: Australian Curriculum. We show specific outcome codes only where they have been verified against the official curriculum document.
What's next?
Completed: Discrete Random Variables and Expected Value
Ready for more? Move on to Year 12 Maths.
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 11 students (typically ages 16–17) working on probability. It is pitched at level 3 · challenge; an easier version is linked above.
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 45 minutes for 15 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?
Explore more Year 11 mathematics resources.
Preview
Success TutoringWorksheetsYear 11 · Mathematics · ProbabilityDiscrete Random Variables and Expected Value
Name
Date
Score /
Show your working. Give decimal answers exactly or to 2 decimal places unless told otherwise.
Part A: Probability distributions
A random variable X assigns a number to each outcome of a random experiment. It is discrete if it takes separate (countable) values, such as the number of sixes in four rolls. Its probability distribution lists each value x with its probability P(X = x).
In a valid distribution, every probability is between 0 and 1 and the probabilities add to 1.
Worked example. X is the number shown on a spinner with P(X = 1) = 0.5, P(X = 2) = 0.3 and P(X = 3) = c. Then 0.5 + 0.3 + c = 1, so c = 0.2, and P(X > 1) = 0.3 + 0.2 = 0.5.
1.The random variable X has P(X = 0) = 0.1, P(X = 1) = 0.3, P(X = 2) = k and P(X = 3) = 0.2. Find k. (answer space)
2.For the distribution in question 1, find P(X ≥ 2). (answer space)
3.A table with outcomes 1, 2, 3 and probabilities 0.2, 0.5 and 0.4 is a valid probability distribution. ☐ True ☐ False
4.Which of these is a discrete random variable?
- AThe number of heads when a coin is tossed 10 times
- BThe time taken to run 100 m
- CThe mass of a randomly chosen apple
- DThe height of a randomly chosen student
Part B: Expected value
The expected value (mean) of a discrete random variable is E(X) = μ = Σ x P(X = x): multiply each value by its probability and add. It is the long-run average value of X over many repeats.
A game is fair when the expected gain (expected winnings − cost to play) is 0.
Worked example. A spinner pays $0 with probability 0.5, $2 with probability 0.3 and $10 with probability 0.2. E(payout) = 0 × 0.5 + 2 × 0.3 + 10 × 0.2 = 0.6 + 2 = $2.60. If a spin costs $3, the expected gain is 2.60 − 3 = −$0.40.
5.Using the distribution in question 1 (P(X = 0) = 0.1, P(X = 1) = 0.3, P(X = 2) = 0.4, P(X = 3) = 0.2), find E(X).
6.A fair six-sided die is rolled. Find the expected value of the number shown. (answer space)
7.It costs $5 to play a game. A fair die is rolled: a 6 wins $20, a 4 or a 5 wins $5, and any other number wins nothing. Find the expected winnings and decide whether the game is fair.
8.A raffle sells 500 tickets at $4 each. There is one prize of $1000 and two prizes of $250. Find the expected gain for a person who buys one ticket.
9.A game has an expected gain of −$0.50 per play for the player. Which statement is correct?
- AOver many games, a player loses about 50 cents per game on average
- BA player loses exactly 50 cents every game
- CThe game is fair
- DA player cannot win any money
Part C: Variance and standard deviation
The variance measures spread about the mean: Var(X) = E(X[sup 2]) − [E(X)][sup 2], where E(X2) = Σ x2 P(X = x). The standard deviation is SD(X) = √Var(X), in the same units as X.
For a linear change Y = aX + b: E(aX + b) = aE(X) + b and Var(aX + b) = a[sup 2] Var(X), so SD(aX + b) = |a| SD(X).
Worked example. X takes values 1 and 3, each with probability 0.5. E(X) = 2, E(X2) = 0.5 × 1 + 0.5 × 9 = 5, so Var(X) = 5 − 22 = 1 and SD = 1.
10.For the distribution in question 1, find Var(X) and the standard deviation of X.
11.A random variable has E(X) = 4 and Var(X) = 2.25. Find E(2X + 3), Var(2X + 3) and the standard deviation of 2X + 3.
12.For constants a and b, Var(aX + b) = (answer space) × Var(X).
Preview — the PDF contains the full resource and answer sheet.
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Probability in other year levels
- Year 9 Probability Worksheet — Level 1
Worksheet · Year 9 · Probability · Level 1 · Easier
- Year 9 Probability Worksheet — Level 2
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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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