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IB Mathematics: Applications and Interpretation SL · 13.2: Probability Distributions · 13.2.3

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Fair games and further distributions Problem set B

Use E(X) = Σ x P(X = x). For a game, expected gain = expected payout − stake. Give decimals exactly or to 3 significant figures.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. You pay £2.50 to play a game. A fair four-sided die (faces 1 to 4) is rolled and you receive the score in pounds.

    1. (a) Find the expected payout.
    2. (b) Is the game fair? Give a reason.

  2. A discrete random variable X has \(P(X=x)=\frac{x^2+2}{22}\) for x = 0, 1, 2, 3.

    1. (a) Write the probability distribution of X as a table.
    2. (b) Find E(X).

  3. A fair four-sided die (faces 1 to 4) is rolled and a fair five-sided spinner (sectors 1 to 5) is spun. X is the sum of the two scores. The table shows every outcome.

    +12345123456234567345678456789
    1. (a) Write the probability distribution of X as a table.
    2. (b) Find the expected total score.

  4. A basketball player scores (S) a free throw with probability 0.7. She takes three throws. X is the number of throws scored. The tree diagram shows the outcomes.

    0.7S0.7S0.7SSSS0.3MSSM0.3M0.7SSMS0.3MSMM0.3M0.7S0.7SMSS0.3MMSM0.3M0.7SMMS0.3MMMM
    1. (a) Find the probability of exactly two throws scored.
    2. (b) Find the probability distribution of X.
    3. (c) Find E(X).

  5. Two names are drawn from a hat of 10 names without replacement. 3 of the names are Year 12 students. X is the number of Year 12 names drawn.

    1. (a) Find the probability distribution of X.
    2. (b) Find E(X).

  6. At a fair, two spinners are each numbered 1 to 5 and spun together. The score X is the positive difference between the two numbers. The table shows every outcome.

    gap12345101234210123321012432101543210
    1. (a) Write the probability distribution of X as a table.
    2. (b) Find the expected score E(X).

In context

  1. At a summer fair, 100 raffle tickets are sold at $1 each. There is one prize of $40 and two prizes of $10.

    1. (a) Find the expected prize for one ticket.
    2. (b) Find the expected gain for a ticket buyer. Is the raffle fair?
    3. (c) Find the organiser's total profit if all the tickets are sold.

  2. A player rolls a fair six-sided die. If it shows a 6 the player wins $15. Otherwise the player pays $n. Find the value of n that makes the game fair.

  3. The number of absent students, X, in a class on a given day has \(P(X=x)=k(6-x)\) for x = 0, 1, 2, 3, 4, where k is a constant.

    1. (a) Find the value of k.
    2. (b) Find P(X ≤ 1).
    3. (c) Find E(X).

Stretch

  1. The table shows the probability distribution of X. The expected value of X is 1.7.

    x0123P(X = x)0.2ab0.25
    1. (a) Write down two equations in a and b.
    2. (b) Find the values of a and b.

  2. A discrete random variable X has \(P(X=x)=cx^2\) for x = 2, 3, 4, 5, where c is a constant.

    1. (a) Find the value of c.
    2. (b) Find P(X ≥ 4 | X ≥ 3).
    3. (c) Find E(X).

jmaths.xyz · AI SL · 13.2.3 Fair games and further distributions