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IB Mathematics: Applications and Interpretation SL · 13.2: Probability Distributions · 13.2.4

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The binomial distribution Problem set B

State the distribution, then use the calculator's binomial pdf or cdf. Give probabilities 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. For each situation, decide whether X can be modelled by a binomial distribution. If it can, write X ~ B(n, p). If it cannot, say which condition fails.

    1. (a) A fair ten-sided die, numbered 1 to 10, is rolled 20 times. X is the number of rolls that show a multiple of 3.
    2. (b) A player takes free throws until she misses for the first time. X is the number of throws she takes.
    3. (c) A bag holds 10 tiles, 3 of them blue. Four tiles are taken out and not put back. X is the number of blue tiles taken.
    4. (d) In a large city, 18% of cars are silver. 40 cars pass a camera. X is the number of silver cars.

  2. \(X \sim B(19, 0.55)\). Find

    1. (a) \(P(X = 11)\)
    2. (b) \(P(X \le 9)\)
    3. (c) \(P(X > 12)\)
    4. (d) \(P(6 < X \le 10)\)

  3. \(X \sim B(40, 0.07)\). Find

    1. (a) \(P(X < 5)\)
    2. (b) \(P(X \ge 3)\)

In context

  1. Each bus on a commuter's route is late with probability 0.11, independently of the other buses. She takes 18 buses in a month. X is the number of late buses.

    1. (a) Write down the distribution of X.
    2. (b) Find the probability that exactly 3 buses are late.
    3. (c) Find the probability that at least 4 buses are late.
    4. (d) Find the probability that between 2 and 5 buses are late, inclusive.

  2. A class has 9 girls and 11 boys. A teacher chooses 5 pupils at random for a team, one after another, with no pupil chosen twice. X is the number of girls chosen.

    1. (a) Explain why X is not binomial.
    2. (b) Find the probability that the second pupil chosen is a girl, given that the first pupil was a girl. Compare it with the probability that the first pupil is a girl.

Stretch

  1. Each person stopped in the street independently gives to a charity with probability 0.25. 30 people are stopped. X is the number who give.

    1. (a) Find the probability that at least 4 but fewer than 9 people give.
    2. (b) Find the probability that at least 10 people give.

  2. 12% of the pupils in a school have flu. The flu spreads easily between pupils who share a room. 10 pupils from one room are chosen. X is the number with flu. Explain why X may not be binomial.

jmaths.xyz · AI SL · 13.2.4 The binomial distribution