Probability distributions: exam review Exam review

The discrete random variable X has the probability distribution shown in the table, where k is a constant. [5]

1. (a) Find the value of k. [2]
2. (b) Find \(P(X \geq 4)\). [1]
3. (c) Find \(P(X \leq 3 \mid X \geq 2)\). [2]

The discrete random variable X has probability distribution \(P(X = x) = k(5 - x)\) for \(x = 1, 2, 3, 4\), where k is a constant. [4]

1. (a) Find the value of k. [2]
2. (b) Find \(P(X \geq 2 \mid X \leq 3)\). [2]

A fair six-sided die is rolled in a game at a fair. A score of 6 wins a prize of \$12. A score of 4 or 5 wins a prize of \$3. Any other score wins nothing. Let X be the prize in dollars. [5]

1. (a) Find the expected prize. [2]
2. (b) The game costs \$4 to play. Find the expected gain of a player and state whether the game is fair. [2]
3. (c) Find the cost to play that would make the game fair. [1]

A bag holds 4 red sweets and 5 white sweets. Two sweets are taken at random, without replacement. Let X be the number of red sweets taken. [6]

1. (a) Find the probability distribution of X, as a table. [3]
2. (b) Find E(X). [2]
3. (c) Find \(P(X \geq 1)\). [1]

Each bulb made by a factory is faulty with probability 0.08, independently of the others. A box holds 20 bulbs. Let X be the number of faulty bulbs in a box. [6]

1. (a) Write down the distribution of X in the form \(X \sim B(n, p)\). [1]
2. (b) Find the probability that exactly 2 bulbs are faulty. [2]
3. (c) Find the probability that at most 3 bulbs are faulty. [1]
4. (d) Find the probability that at least 3 bulbs are faulty. [2]

Some random variables are modelled by binomial distributions. [7]

1. (a) Find E(X) and Var(X) for \(X \sim B(40, 0.35)\). [2]
2. (b) \(Y \sim B(n, p)\) has mean 18 and variance 12.6. Find the value of p and the value of n. [3]
3. (c) Find \(P(Y > 18)\). [2]

A multiple-choice test has 10 questions. Each question has 4 answers, only one of which is correct. A student passes by answering at least 5 questions correctly. Mia guesses the answer to every question. [6]

1. (a) State two assumptions that are needed to model the number of correct answers as a binomial distribution. [2]
2. (b) Find the probability that a student who guesses every answer passes. [2]
3. (c) Twelve students each guess every answer. Assuming each student passes independently, find the probability that exactly 2 students pass. [2]

A factory packs light bulbs in boxes. Each bulb is faulty with probability 0.05, independently of the others. [8]

1. (a) A box contains 12 bulbs. Find the probability that at least one bulb in the box is faulty. [2]
2. (b) A larger box contains n bulbs. Find the smallest value of n for which the probability that at least one bulb is faulty is greater than 0.95. [3]
3. (c) An airline sells n tickets for a flight with 60 seats. Each passenger who buys a ticket turns up with probability 0.92, independently of the others. Find the largest value of n for which the probability that more passengers turn up than there are seats is less than 0.05. [3]

The time that a player takes to finish a puzzle is normally distributed with mean 240 seconds and variance 225 seconds². [4]

1. (a) Use the 68–95–99.7 rule to estimate the percentage of player times between 210 and 255 seconds. [2]
2. (b) Use your calculator to find the probability that a player's time is longer than 270 seconds. [2]

The masses of eggs laid on a farm are normally distributed with mean 58 g and standard deviation 3.2 g. [9]

1. (a) Find the symmetric interval about the mean that contains the masses of the middle 90% of the eggs. [3]
2. (b) Find the lower quartile and the upper quartile of the masses. [2]
3. (c) The masses of eggs from a second farm are normally distributed with mean \(\mu\) g and standard deviation \(\sigma\) g. On this farm 10% of the eggs have mass less than 54 g and 20% have mass greater than 62 g. Find the value of \(\mu\) and the value of \(\sigma\). [4]

The mass of a loaf of bread made by a bakery is normally distributed with mean 500 g and variance 144 g². The diagram shows the distribution, with the loaves lighter than 484 g shaded. [14]

1. (a) Find the probability that a loaf weighs less than 484 g. This is the shaded area in the diagram. [2]
2. (b) Find the probability that a loaf is between 480 and 510 g. [2]
3. (c) The bakery rejects the lightest 4% of loaves. Find the least mass of a loaf that is not rejected. [2]
4. (d) The bakery makes 800 loaves in a day. Find the expected number that weigh more than 520 g. [2]
5. (e) Ten loaves are chosen at random. Find the probability that exactly 2 loaves weigh less than 484 g. [3]
6. (f) State one assumption that you made in part (e). [1]
7. (g) The bakery also makes small rolls. Their mass has mean 40 g and standard deviation 22 g. Explain why a normal distribution is not a good model for the mass of a roll. [2]

At a fair, a player spins a spinner and receives X dollars, where X is the score. The probability distribution of X is shown in the table. [13]

1. (a) Write down an equation connecting a and b. [1]
2. (b) Given that E(X) = 2.75, find the value of a and the value of b. [3]
3. (c) Find \(P(X \geq 3 \mid X \geq 2)\). [2]
4. (d) A spin costs \$3. Find the expected profit of a player and state whether the game is fair. [2]
5. (e) The spinner is spun 8 times. Find the probability that exactly 3 of the 8 scores are 4. [3]
6. (f) Find the probability that at least 2 of the 8 scores are 4. [2]

The time that a courier takes to make a delivery, in minutes, is modelled by \(X \sim N(42, 8^2)\). [13]

1. (a) The box plot shows the times of a random sample of 200 deliveries. Give two features of the box plot that suggest a normal distribution is a suitable model. [2]
2. (b) Find the lower quartile and the upper quartile of X. [2]
3. (c) Hence find the interquartile range. [1]
4. (d) The slowest 6% of deliveries are called very late. Find the time that separates these from the others. [2]
5. (e) Find the value of k such that \(P(42 < X < k) = 0.4\). [3]
6. (f) The courier is paid \$6 for a delivery that takes less than 35 minutes, \$4 for a delivery that takes between 35 and 50 minutes and \$2 for a delivery that takes more than 50 minutes. Find the expected pay for one delivery. [3]