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IB Mathematics: Applications and Interpretation SL · 13.2: Probability Distributions · 13.2.10

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Distributions review Problem set B

Mixed questions on discrete, binomial and normal distributions. Show the distribution you use and give 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. The random variable X is the number of goals a team scores in a match. The table shows its probability distribution.

    x0123P(X = x)0.082k0.2k
    1. (a) Find the value of k.
    2. (b) Find E(X).
    3. (c) Find P(X ≥ 2).

  2. A student guesses the answers to 60 multiple-choice questions. Each guess is correct with probability 0.25. Let Y be the number correct, so Y follows a binomial distribution. Write down E(Y).

  3. A light bulb is faulty with probability 0.15. Twenty bulbs are tested. Let Y be the number that are faulty, so \(Y \sim B(20, 0.15)\).

    1. (a) Find the probability of exactly 4.
    2. (b) Find the probability of at least 4.

  4. The time a runner takes to complete a 10 km race is normally distributed with mean 55 minutes and standard deviation 6 minutes. Find the probability that a runner takes between 50 and 62 minutes.

In context

  1. The mass of a loaf is normally distributed with mean 820 g and standard deviation 15 g. A bakery makes 300 loaves in a day.

    1. (a) Find the probability that a loaf has mass between 800 g and 850 g.
    2. (b) Find the expected number of loaves in a day with mass less than 795 g.

  2. The score of a student in a physics test is normally distributed with mean 61 and standard deviation 11. The top 15% of students get a distinction. Find the lowest score that gets a distinction.

Stretch

  1. The time a delivery takes is normally distributed with mean 36 hours and standard deviation 5 hours. A delivery is late if it takes more than 45 hours. Of 10 deliveries, let Y be the number that are late.

    1. (a) Find the probability that one delivery is late.
    2. (b) Find P(Y = 1).

  2. An archer hits the target with probability 0.8. She shoots 12 arrows, independently. Let Y be the number of hits.

    1. (a) Write down the expected number of hits.
    2. (b) Find the probability that exactly 10 are hits.
    3. (c) Find the probability that at least 10 are hits.

jmaths.xyz · AI SL · 13.2.10 Distributions review: chapter 8 checklist and short quiz