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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 A

Mixed questions on discrete, binomial and normal distributions. Show the distribution you use and give 3 significant figures.

Difficulty: first skills applying them stretch

Skills

  1. The random variable X is the score when a spinner is spun. The table shows its probability distribution.

    x1234P(X = x)0.16k0.323k
    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 40 true-or-false questions. Each guess is correct with probability 0.5. Let Y be the number correct, so Y follows a binomial distribution. Write down E(Y).

  3. A seed germinates with probability 0.35. Twelve seeds are planted. Let Y be the number that germinate, so \(Y \sim B(12, 0.35)\).

    1. (a) Find the probability of exactly 5.
    2. (b) Find the probability of at least 5.

  4. The mass of a pear is normally distributed with mean 160 g and standard deviation 12 g. Find the probability that a pear has mass between 150 g and 175 g.

In context

  1. The time a bus takes to complete a route is normally distributed with mean 42 minutes and standard deviation 5 minutes. A day has 80 journeys.

    1. (a) Find the probability that a journey takes between 38 and 50 minutes.
    2. (b) Find the expected number of journeys in a day that take less than 35 minutes.

  2. The height of a species of tree is normally distributed with mean 18.0 m and standard deviation 2.5 m. The tallest 12% of trees are marked for study. Find the least height of a marked tree.

Stretch

  1. The journey time to school is normally distributed with mean 25 minutes and standard deviation 4 minutes. A journey is late if it takes more than 32 minutes. Over 8 days, let Y be the number of late journeys.

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

  2. A basketball player scores a free throw with probability 0.7. She takes 15 free throws, independently. Let Y be the number she scores.

    1. (a) Write down the expected number of free throws scored.
    2. (b) Find the probability that exactly 12 are scored.
    3. (c) Find the probability that at least 12 are scored.

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