Practice Maths

★ Topic Review — Probability

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This review covers all three lessons: Events, Sets and Venn Diagrams; Probability Rules; and Conditional Probability and Independence. Questions increase in difficulty.

  1. Fluency — Set notation and Venn diagrams

    In a class of 25 students, 15 play sport (S), 12 play a musical instrument (M), and 7 do both.

    1. (a) Draw a Venn diagram and fill in the count in each region.
    2. (b) Find P(S only).
    3. (c) Find P(S ∪ M).
    4. (d) Find P(S′ ∩ M′) (neither).

  2. Fluency — Complement rule

    1. (a) P(A) = 0.38. Find P(A′).
    2. (b) The probability of NOT winning a prize is 7/8. Find P(winning a prize).
    3. (c) P(A′) = 0.45. P(B) = 0.30. A and B are mutually exclusive. Find P(A′ ∩ B′).

  3. Fluency — Addition rule

    1. (a) P(A) = 0.55, P(B) = 0.40, P(A ∩ B) = 0.20. Find P(A ∪ B).
    2. (b) P(A ∪ B) = 0.80, P(A) = 0.50, P(B) = 0.45. Find P(A ∩ B).
    3. (c) A and B are mutually exclusive with P(A) = 2P(B) and P(A ∪ B) = 0.6. Find P(A) and P(B).

  4. Fluency — Two-way table

    150 people surveyed. Rows: Employed(E)/Unemployed(U). Columns: City(C)/Regional(R). E/C=54, E/R=36, U/C=30, U/R=30. Total=150.

    1. (a) Find P(Employed).
    2. (b) Find P(City ∩ Unemployed).
    3. (c) Find P(Employed ∪ City).

  5. Fluency — Conditional probability basics

    1. (a) P(A ∩ B) = 0.18, P(B) = 0.6. Find P(A|B).
    2. (b) P(A|B) = 0.4, P(B) = 0.35. Find P(A ∩ B).
    3. (c) A bag has 6 red and 4 blue marbles. One red marble is removed. What is the probability the next marble drawn is red?

  6. Understanding — Three-set Venn diagram

    80 students surveyed about social media use. Facebook(F): 45, Instagram(I): 38, Twitter(T): 22. F∩I: 18, F∩T: 12, I∩T: 10. F∩I∩T: 5.

    1. (a) Find n(F ∪ I ∪ T).
    2. (b) Find the number using none of the three.
    3. (c) Find P(Facebook only).
    4. (d) Find P(exactly two platforms).

  7. Understanding — Testing independence

    A survey finds: P(Owns a car) = 0.6, P(Has a full-time job) = 0.7, P(Owns a car ∩ Full-time job) = 0.42.

    1. (a) Are “Owns a car” and “Full-time job” independent? Show your working.
    2. (b) Find P(Owns a car | Full-time job).
    3. (c) Find P(Full-time job | Owns a car).
    4. (d) Interpret the independence result in context.

  8. Understanding — Tree diagram

    A box has 5 green and 3 yellow balls. Two are drawn without replacement.

    1. (a) Draw a tree diagram with all probabilities on each branch.
    2. (b) Find P(same colour both draws).
    3. (c) Find P(at least one green).
    4. (d) Find P(1st is green | 2nd is yellow).

  9. Problem Solving — Medical screening

    Challenge. 2% of athletes use a prohibited substance (D). A drug test has P(positive|D)=0.98 and P(negative|D′)=0.95. An athlete tests positive.
    1. (a) Using a 10 000-person table, find P(positive test).
    2. (b) Find P(D | positive) — the probability the athlete actually used the substance given a positive result.
    3. (c) Should we automatically conclude a positive-testing athlete is guilty? Explain using your answer to (b).

  10. Problem Solving — Combined probability reasoning

    Challenge.
    1. (a) P(A) = 0.5, P(B) = 0.4. A and B are independent. Find P(A′ ∩ B), P(A ∩ B′), and P(A′ ∩ B′). Verify all four regions of the Venn diagram sum to 1.
    2. (b) A and B are mutually exclusive with P(A)=0.3 and P(B)=0.4. C is independent of both A and B with P(C)=0.5. Find P(A ∩ C) and P((A∪B) ∩ C).
    3. (c) Cards are drawn from a standard deck. Find P(3rd card is an Ace | first two cards are not Aces). Show full working.

  11. Problem Solving — Extended Venn diagram

    Challenge. P(A)=0.5, P(B)=0.4, P(C)=0.3. P(A∩B)=0.2, P(A∩C)=0.15, P(B∩C)=0.12, P(A∩B∩C)=0.08.
    1. (a) Find P(A ∪ B ∪ C).
    2. (b) Find the probability of exactly one event occurring.
    3. (c) Find P(A only).
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