Probability distributions: exam review Exam review: second paper

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 \(P(X \geq 3)\). [1]
3. (c) Find \(P(X \leq 2 \mid X \geq 1)\). [2]

The discrete random variable Y has probability distribution \(P(Y = y) = ky\) for \(y = 1, 2, 3, 4\), where k is a constant. [4]

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

A fair spinner has 10 equal sectors numbered 1 to 10. It is spun in a game at a school stall. A score of 10 wins a prize of \$20. A score of 7, 8 or 9 wins a prize of \$5. 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 \$5 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 box holds 3 faulty bulbs and 7 working bulbs. Two bulbs are taken at random, without replacement. Let X be the number of faulty bulbs 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 seed in a tray fails to germinate with probability 0.12, independently of the others. A tray holds 15 seeds. Let X be the number of seeds in a tray that fail to germinate. [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 3 seeds fail to germinate. [2]
3. (c) Find the probability that at most 4 seeds fail to germinate. [1]
4. (d) Find the probability that at least 4 seeds fail to germinate. [2]

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

1. (a) Find E(X) and Var(X) for \(X \sim B(25, 0.6)\). [2]
2. (b) \(Y \sim B(n, p)\) has mean 20 and variance 15. Find the value of p and the value of n. [3]
3. (c) Find \(P(Y > 20)\). [2]

A quiz has 12 questions. Each question has 5 answers, only one of which is correct. A student passes by answering at least 5 questions correctly. Omar 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) Fifteen students each guess every answer. Assuming each student passes independently, find the probability that exactly 2 students pass. [2]

A company packs memory cards in boxes. Each card is faulty with probability 0.03, independently of the others. [8]

1. (a) A box contains 15 cards. Find the probability that at least one card in the box is faulty. [2]
2. (b) A larger box contains n cards. Find the smallest value of n for which the probability that at least one card is faulty is greater than 0.9. [3]
3. (c) A theatre sells n tickets for a show with 80 seats. Each person who buys a ticket turns up with probability 0.9, independently of the others. Find the largest value of n for which the probability that more people turn up than there are seats is less than 0.05. [3]

The life of a type of battery is normally distributed with mean 300 hours and variance 400 hours². [4]

1. (a) Use the 68–95–99.7 rule to estimate the percentage of battery lives between 280 and 360 hours. [2]
2. (b) Use your calculator to find the probability that a battery's life is longer than 330 hours. [2]

The heights of seedlings in a nursery are normally distributed with mean 120 mm and standard deviation 8.5 mm. [9]

1. (a) Find the symmetric interval about the mean that contains the heights of the middle 80% of the seedlings. [3]
2. (b) Find the lower quartile and the upper quartile of the heights. [2]
3. (c) The heights of seedlings in a second nursery are normally distributed with mean \(\mu\) mm and standard deviation \(\sigma\) mm. In this nursery 5% of the seedlings have height less than 100 mm and 15% have height greater than 130 mm. Find the value of \(\mu\) and the value of \(\sigma\). [4]

The time that a bus takes to travel between two towns is normally distributed with mean 35 minutes and variance 25 minutes². The diagram shows the distribution, with the journeys shorter than 30 minutes shaded. [14]

1. (a) Find the probability that a journey takes less than 30 minutes. This is the shaded area in the diagram. [2]
2. (b) Find the probability that a journey is between 32 and 41 minutes. [2]
3. (c) The fastest 8% of journeys win a prize for the driver. Find the greatest time of a journey that wins a prize. [2]
4. (d) The bus makes 600 journeys in a month. Find the expected number that take more than 45 minutes. [2]
5. (e) Twelve journeys are chosen at random. Find the probability that exactly 3 journeys take less than 30 minutes. [3]
6. (f) State one assumption that you made in part (e). [1]
7. (g) A short route has journey times with mean 6 minutes and standard deviation 4 minutes. Explain why a normal distribution is not a good model for the journey time on this route. [2]

At a charity event, a player turns a wheel and receives X dollars, where X is the number the wheel shows. 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) = 2.6, 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 turn costs \$3. Find the expected profit of a player and state whether the game is fair. [2]
5. (e) The wheel is turned 10 times. Find the probability that exactly 4 of the 10 scores are 4. [3]
6. (f) Find the probability that at least 2 of the 10 scores are 4. [2]

The time that a student takes to finish an online puzzle, in minutes, is modelled by \(X \sim N(38, 6^2)\). [13]

1. (a) The box plot shows the times of a random sample of 150 students. 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 8% of students are given an extra hint. Find the time that separates these from the others. [2]
5. (e) Find the value of k such that \(P(38 < X < k) = 0.35\). [3]
6. (f) A student scores 10 points for a time under 30 minutes, 6 points for a time between 30 and 45 minutes and 2 points for a time over 45 minutes. Find the expected score for one student. [3]

The number of customers who arrive at a bicycle repair stall is modelled by a Poisson distribution X with a constant average rate of 7.5 customers per hour. [10]

1. (a) State two conditions that must hold for X to be modelled by a Poisson distribution, in the context of customers arriving at the stall. [2]
2. (b) Find the probability that exactly 9 customers arrive in one hour. [2]
3. (c) Find the probability that more than 10 customers arrive in one hour. [2]
4. (d) Find the probability that at least 3 customers arrive in a period of 20 minutes. [3]
5. (e) Find the standard deviation of X. [1]

Two ticket machines at a station are used independently. The number of tickets per hour sold by machine A is modelled by \(X \sim \text{Po}(6)\). Machine B sells tickets at an average of 5 per 20 minutes, and the number it sells is modelled by a Poisson distribution Y. X and Y are assumed to be independent. [10]

1. (a) Show that the mean number of tickets per hour at machine B is 15. [1]
2. (b) Write down the distribution of Z, the total number of tickets per hour at machine A and machine B together. [2]
3. (c) Find \(P(Z \leq 17)\). [2]
4. (d) Find E(Z) and the standard deviation of Z. [2]
5. (e) Explain why the number of tickets at machine A and the number at machine B may not be independent. [1]
6. (f) Hana says that T, the time in minutes between one of the tickets and the next, follows a Poisson distribution. Explain why she is wrong. [2]

The number of orders that arrive at a bakery counter is modelled by a Poisson distribution with a constant average rate of 2.8 orders per hour. [10]

1. (a) Find the most likely number of orders in one hour. Justify your answer. [3]
2. (b) On a quiet day the number of orders per hour has mean \(\lambda\). The probability that at least one order arrives in an hour is 0.7. Find the value of \(\lambda\). [3]
3. (c) On a Friday the counter is open for 4 hours. Find the smallest number of loaves the bakery must bake so that the probability of running out is at most 0.05, if each order uses one loaf. [4]