Distributions review Problem set B One question at a time

The random variable X is the number of goals a team scores in a match. The table shows its probability distribution.

1. (a) Find the value of k.

   • The probabilities add to 1: 0.08 + 2k + 0.2 + k = 1
   • 0.28 + 3k = 1, so k = 0.24

2. (b) Find E(X).

   • \(E(X) = 0 \times 0.08 + 1 \times 0.48 + 2 \times 0.2 + 3 \times 0.24 = 1.6\)

3. (c) Find P(X ≥ 2).

   • P(X ≥ 2) = 0.2 + 0.24 = 0.44

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).

• \(Y \sim B(60, 0.25)\)
• \(E(Y) = np = 60 \times 0.25 = 15\)

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.

   • \(Y \sim B(20, 0.15)\)
   • From the calculator: binomial pdf with n = 20, p = 0.15, x = 4: 0.182

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

   • \(P(Y \geq 4) = 1 - P(Y \leq 3)\)
   • From the calculator: binomial cdf with n = 20, p = 0.15, upper = 3: 0.6477
   • 1 − 0.6477 = 0.352

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.

• \(X \sim N(55, 6^2)\)
• From the calculator: normal cdf with lower = 50, upper = 62, μ = 55, σ = 6
• P = 0.676

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.

   • From the calculator: normal cdf with lower = 800, upper = 850, μ = 820, σ = 15
   • P = 0.886

2. (b) Find the expected number of loaves in a day with mass less than 795 g.

   • From the calculator: normal cdf with lower = −10^99, upper = 795, μ = 820, σ = 15: 0.04779
   • \(300 \times 0.04779 = 14.3\)
   • About 14

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.

• P(X > k) = 0.15, so P(X < k) = 0.85
• From the calculator: inverse normal with area = 0.85, μ = 61, σ = 11
• k = 72.4

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.

   • From the calculator: normal cdf with lower = 45, upper = 10^99, μ = 36, σ = 5
   • p = 0.0359

2. (b) Find P(Y = 1).

   • \(Y \sim B(10, 0.03593)\)
   • From the calculator: binomial pdf with n = 10, p = 0.03593, x = 1: 0.258

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.

   • \(np = 12 \times 0.8 = 9.6\)

2. (b) Find the probability that exactly 10 are hits.

   • From the calculator: binomial pdf with n = 12, p = 0.8, x = 10: 0.283

3. (c) Find the probability that at least 10 are hits.

   • From the calculator: binomial cdf with n = 12, p = 0.8, upper = 9: 0.4417
   • 1 − 0.4417 = 0.558