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IB Mathematics: Applications and Interpretation HL · 13.2: Probability Distributions · 13.2.6

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Mean, variance and binomial practice Problem set A

Define X and state its distribution. Use E(X) = np and Var(X) = np(1 − p) where asked, and write the calculator settings you used. Give probabilities to 3 significant figures.

Difficulty: first skills applying them stretch

Skills

  1. \(X \sim B(32, 0.35)\). Find

    1. (a) the mean of X
    2. (b) the variance of X
    3. (c) the standard deviation of X

  2. \(Y \sim B(22, 0.35)\). Find

    1. (a) \(P(Y = 8)\)
    2. (b) \(P(Y > 9)\)
    3. (c) \(P(5 \le Y < 9)\)

  3. Each of 6 seedlings independently survives a frost with probability 0.3. X is the number of seedlings that survive.

    x0123456P(X = x)???????
    1. (a) Complete the table of the probability distribution of X. Give each value to 3 significant figures.
    2. (b) Write down the most likely value of X.

In context

  1. A pack holds 8 pens. Each pen is faulty with probability 0.04, independently. A school buys 12 packs and returns any pack that has at least one faulty pen.

    1. (a) Find the probability that a pack has at least one faulty pen.
    2. (b) Find the probability that the school returns more than 2 of the 12 packs.

  2. A quiz has 28 questions, each with 4 possible answers. Priya guesses every answer at random. X is the number of questions she gets correct.

    1. (a) Find the probability that Priya gets fewer than 5 correct.
    2. (b) Find the expected number of correct answers.
    3. (c) Find the probability that she gets more than the expected number correct.

  3. A courier says that 4% of the parcels it delivers arrive damaged. A van carries 60 parcels. X is the number of damaged parcels.

    1. (a) State two assumptions needed for X to be modelled by a binomial distribution.
    2. (b) Suggest why one of these assumptions might not hold.
    3. (c) Assume the model is valid. Find the probability that exactly 3 parcels are damaged.

  4. A packet of cereal contains a free toy with probability 0.06, independently of other packets. Sam buys packets one after another.

    1. (a) Find the probability that at least one of the next 10 has the prize.
    2. (b) Find the smallest number needed so that the probability of at least one prize is greater than 0.95.

Stretch

  1. A biased spinner lands on red with probability p. In 15 spins the mean number of reds is 9. X is the number of reds in 15 spins.

    1. (a) Find the value of p.
    2. (b) Find the probability that the next 15 spins give fewer than 6 reds.

  2. X ~ B(n, p). The mean of X is 20 and the standard deviation of X is √12.

    1. (a) Find n and p.
    2. (b) Find the probability that X takes the value of its mean.

  3. A ferry has room for 50 cars. Each driver who books turns up independently with probability 0.9. The company accepts n bookings and X is the number of cars that arrive.

    1. (a) Find the mean number who arrive if 55 are sold.
    2. (b) Find the largest number that can be sold so that the probability that more than 50 arrive is less than 0.05.

jmaths.xyz · AI HL · 13.2.6 Mean, variance and binomial practice