Practice Maths

T3T2 Review — Probability

Mixed questions covering Introduction to Probability (L31), Conditional Probability and Independence (L32), and Probability Distributions (L33).

  1. Basic probability. Fluency

    A bag contains 5 red, 3 blue and 2 green marbles.

    • (a) Find P(red).
    • (b) Find P(not blue).
    • (c) One marble is drawn. Find P(red or green).
    • (d) Are drawing a red and drawing a blue marble mutually exclusive? Explain.

  2. Addition rule. Fluency

    In a class of 30 students, 18 study French (F), 12 study Spanish (S), and 6 study both.

    • (a) Find P(F).
    • (b) Find P(F ∪ S) using the addition rule.
    • (c) Find P(neither French nor Spanish).
    • (d) Find P(F only, not S).

  3. Conditional probability. Fluency

    • (a) P(A ∩ B) = 0.15, P(B) = 0.5. Find P(A|B).
    • (b) P(A) = 0.4, P(B) = 0.6, P(A ∩ B) = 0.24. Are A and B independent?
    • (c) A card is drawn from a standard deck. Find P(Ace | red card).
    • (d) A card is drawn. Find P(red | Ace).

  4. Probability distributions. Fluency

    • (a) X takes values 1, 2, 3 with P = 0.3, 0.5, k. Find k.
    • (b) Using the distribution from (a), find E(X).
    • (c) X takes values 0, 1, 2 with P = 1/3, 1/2, k. Find k.
    • (d) Two dice are rolled. Let X = 1 if a double is rolled, X = 0 otherwise. Find E(X).

  5. Two-way table — exercise habits. Understanding

    A survey of 200 people asked whether they exercise regularly (E) and whether they have a healthy diet (D). The results are shown.

    Diet (D) No Diet (D') Total Exercise (E) No Exercise (E') 90 30 120 30 50 80
    Survey of 200 people (total row not shown — calculate it)
    • (a) Find P(E ∩ D).
    • (b) Find P(D | E).
    • (c) Find P(E | D).
    • (d) Are E and D independent? Show working.

  6. Drawing without replacement. Understanding

    A box has 4 yellow and 6 purple marbles. Two marbles are drawn without replacement.

    • (a) Find P(both yellow).
    • (b) Find P(first yellow, second purple).
    • (c) Find P(one of each colour).
    • (d) Find P(at least one purple).

  7. Tree diagram — two-stage experiment. Understanding

    A student takes a maths test (P(pass) = 0.7) and a science test (P(pass) = 0.6) independently.

    • (a) Find P(pass both).
    • (b) Find P(pass exactly one).
    • (c) Find P(fail both).
    • (d) Given the student passed maths, find P(pass science).

  8. Expected value in context. Understanding

    X is the number of goals a team scores per game. The distribution is: P(0) = 0.20, P(1) = 0.35, P(2) = 0.30, P(3) = 0.10, P(4) = 0.05.

    • (a) Verify this is a valid distribution.
    • (b) Find E(X).
    • (c) Find P(X ≥ 2).
    • (d) In a season of 20 games, how many goals would be expected in total?

  9. Bayes’ theorem — testing for defects. Problem Solving

    A factory produces items. 5% are defective. A quality-control test has 90% accuracy for defective items (true positive) and 5% false positive rate.

    • (a) Define events D (defective) and T (test positive). State P(D), P(T|D), P(T|D′).
    • (b) Find P(T) using the law of total probability.
    • (c) Find P(D|T) using Bayes’ theorem.
    • (d) Interpret the result: if an item tests positive, what is the probability it is actually defective?

  10. Expected value — game design. Problem Solving

    A charity raffle has 1 000 tickets. Prizes: 1 × $500, 2 × $100, 5 × $20.

    • (a) Write the probability distribution of prize X for one ticket.
    • (b) Find E(X).
    • (c) If tickets cost $2 each, find the expected total profit for the charity.
    • (d) Find the ticket price that would make E(profit per ticket) = $1.20 for the charity.

  11. Multi-step conditional probability. Problem Solving

    Three machines A, B, C produce 50%, 30%, 20% of a factory’s output. Defect rates: A = 2%, B = 3%, C = 5%.

    • (a) Find P(defective and from A), P(defective and from B), P(defective and from C).
    • (b) Find P(defective).
    • (c) Given a defective item, find P(it came from machine C).
    • (d) Which machine is most likely to have produced a randomly chosen defective item?

  12. Distribution meets conditional probability. Problem Solving

    A game show host picks a door at random from 3 doors; one hides a car, two hide goats. The contestant also picks a door at random. Let X = 1 if the contestant wins the car, X = 0 otherwise.

    • (a) Find P(X = 1).
    • (b) Find E(X).
    • (c) The host (who knows where the car is) opens a goat door that the contestant did not pick. The contestant now has the option to switch. By listing the cases, show that switching gives P(win) = 2/3.
    • (d) If the game is played 90 times and the contestant always switches, what is the expected number of cars won?
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