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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 B

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.

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. \(X \sim B(45, 0.2)\). Find

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

  2. \(Y \sim B(26, 0.4)\). Find

    1. (a) \(P(Y = 11)\)
    2. (b) \(P(Y > 12)\)
    3. (c) \(P(8 \le Y < 12)\)

  3. Each of 6 pupils in a club independently walks to school with probability 0.65. X is the number of pupils who walk to school.

    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 carton holds 9 yoghurt pots. Each pot is damaged with probability 0.03, independently. A cafe orders 14 cartons and returns any carton that has at least one damaged pot.

    1. (a) Find the probability that a carton has at least one damaged pot.
    2. (b) Find the probability that the cafe returns more than 3 of the 14 cartons.

  2. A quiz has 40 questions, each with 5 possible answers. Omar guesses every answer at random. X is the number of questions he gets correct.

    1. (a) Find the probability that Omar gets fewer than 6 correct.
    2. (b) Find the expected number of correct answers.
    3. (c) Find the probability that he gets more than the expected number correct.

  3. A laboratory finds that 3% of the test tubes it puts in an oven crack. One oven load holds 50 tubes. X is the number of tubes that crack in one load.

    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 2 tubes crack.

  4. An arcade grab machine gives a prize on each try with probability 0.04, independently of other tries. Jo plays the machine again and again.

    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 0.9.

Stretch

  1. A basketball machine is set so that each shot scores with probability p. In 20 shots the mean number of scores is 7. X is the number of scores in 20 shots.

    1. (a) Find the value of p.
    2. (b) Find the probability that the next 20 shots give more than 9 scores.

  2. X ~ B(n, p). The mean of X is 27 and the standard deviation of X is √18.9.

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

  3. A theatre has 120 seats. Each person who books a seat turns up independently with probability 0.95. The theatre accepts n bookings and X is the number of people who arrive.

    1. (a) Find the mean number who arrive if 126 are sold.
    2. (b) Find the largest number that can be sold so that the probability that more than 120 arrive is less than 0.05.

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