5.02a Discrete probability distributions: general

Four letters are taken out of their envelopes for signing. Unfortunately they are replaced randomly, one in each envelope. The probability distribution for the number of letters, \(X\), which are now in the correct envelope is given in the following table.

\(r\)	0	1
P(X = r)	\(\frac{3}{8}\)	\(\frac{1}{3}\)	\(\frac{1}{4}\)	0	\(\frac{1}{24}\)

1. Explain why the case \(X = 3\) is impossible. [1]
2. Explain why P(\(X = 4\)) = \(\frac{1}{24}\). [2]
3. Calculate E(\(X\)) and Var(\(X\)). [5]

Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution P(\(X = r\)) = \(\frac{1}{40}r(r + 1)\) for \(r = 1, 2, 3, 4\).

1. Verify that P(\(X = 4\)) = \(\frac{1}{2}\). [1]
2. Calculate E(\(X\)) and Var(\(X\)). [5]
3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days. [2]

The number, \(X\), of children per family in a certain city is modelled by the probability distribution P(\(X = r\)) = \(k(6 - r)(1 + r)\) for \(r = 0, 1, 2, 3, 4\).

1. Copy and complete the following table and hence show that the value of \(k\) is \(\frac{1}{50}\). [3]

   \(r\)	0	1	2	3	4
   P(\(X = r\))	\(6k\)	\(10k\)

2. Calculate E(\(X\)). [2]
3. Hence write down the probability that a randomly selected family in this city has more than the mean number of children. [1]

Members of a library may borrow up to 6 books. Past experience has shown that the number of books borrowed, \(X\), follows the distribution shown in the table.

\(x\)	0	1	2	3	4	5	6
P(X = x)	0	0.19	0.26	0.20	0.13	0.07	0.15

1. Find the probability that a member borrows more than 3 books. [1 mark]
2. Assume that the numbers of books borrowed by two particular members are independent. Find the probability that one of these members borrows more than 3 books and the other borrows fewer than 3 books. [3 marks]
3. Show that the mean of \(X\) is 3.08, and calculate the variance of \(X\). [4 marks]
4. One of the library staff notices that the values of the mean and the variance of \(X\) are similar and suggests that a Poisson distribution could be used to model \(X\). Without further calculations, give two reasons why a Poisson distribution would not be suitable to model \(X\). [2 marks]
5. The library introduces a fee of 10 pence for each book borrowed. Assuming that the probabilities do not change, calculate:
   1. the mean amount that will be paid by a member;
   2. the standard deviation of the amount that will be paid by a member.
   [3 marks]

The discrete random variable \(X\) has the following probability distribution function. $$\mathrm{P}(X = x) = \begin{cases} 0.2 & x = 1 \\ 0.3 & x = 2 \\ 0.1 & x = 3, 4 \\ 0.25 & x = 5 \\ 0.05 & x = 6 \\ 0 & \text{otherwise} \end{cases}$$ Find the mode of \(X\). Circle your answer. [1 mark] 0.1 \quad 0.25 \quad 2 \quad 3

The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.

\(x\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(P(X = x)\)	\(0.05\)	\(0.2\)	\(0.5\)	\(0.2\)	\(0.05\)

1. 1. Explain why \(\text{E}(X) = 2\). [1]
   2. Find \(\text{Var}(X)\). [3]
2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250X - 80\). Find

The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.

1. Explain why it would be inappropriate to test all the remote controls in this way. [1]
2. State an advantage of using random sampling in this context. [1]

A website awards a random number of loyalty points each time a shopper buys from it. The shopper gets a whole number of points between \(0\) and \(10\) (inclusive). Each possibility is equally likely, each time the shopper buys from the website. Awards of points are independent of each other.

1. Let \(X\) be the number of points gained after shopping once. Find
2. Let \(Y\) be the number of points gained after shopping twice. Find
3. Find the probability of the most likely number of points gained after shopping twice. Justify your answer. [4]

At a bird feeding station, birds are captured and ringed. If a bird is recaptured, the ring enables it to be identified. The table below shows the number of recaptures, \(x\), during a period of a month, for each bird of a particular species in a random sample of \(40\) birds.

Number of recaptures, \(x\)	0	1	2	3	4	5	6	7	8	9	10
Frequency	2	5	5	9	10	4	3	1	0	1	0

1. The sample mean of \(x\) is \(3.4\). Calculate the sample variance of \(x\). [2]
2. Briefly comment on whether the results of part (i) support a suggestion that a Poisson model might be a good fit to the data. [1]

The screenshot below shows part of a spreadsheet for a \(\chi^2\) test to assess the goodness of fit of a Poisson model. The sample mean of \(3.4\) has been used as an estimate of the Poisson parameter. Some values in the spreadsheet have been deliberately omitted. \includegraphics{figure_2}

1. State the null and alternative hypotheses for the test. [1]
2. Calculate the missing values in cells
3. Complete the test at the \(10\%\) significance level. [5]
4. The screenshot below shows part of a spreadsheet for a \(\chi^2\) test for a different species of bird. Find the value of the Poisson parameter used. \includegraphics{figure_3} [3]

A game at a school fete is played with a fair coin and a random number generator which generates random integers between 1 and 52 inclusive. It costs 50 pence to play the game. First, the player tosses the coin. If it lands on tails, the player loses. If it lands on heads, the player is allowed to generate a random number. If the number is 1, the player wins £5. If the number is between 2 and 13 inclusive, the player wins £1. If the number is greater than 13, the player loses.

1. Find the probability distribution of the player's profit. [5]
2. Find the mean and standard deviation of the player's profit. [4]
3. Given that 200 people play the game, calculate
   1. the expected number of players who win some money,
   2. the expected profit for the fete. [2]

A spinner has edges numbered 1, 2, 3, 4 and 5. When the spinner is spun, the number of the edge on which it lands is the score. The probability distribution of the score, \(N\), is given in the table. It is known that E\((N) = 2.55\).

1. Find Var\((N)\). [7]
2. Find E\((3N + 2)\). [1]
3. Find Var\((3N + 2)\). [1]

The discrete random variable \(D\) has the following probability distribution where \(k\) is a constant.

1. Show that the value of \(k\) is \(\frac{600}{137}\) [2]
2. The random variables \(D_1\) and \(D_2\) are independent and each have the same distribution as \(D\). Find \(P(D_1 + D_2 = 80)\) Give your answer to 3 significant figures. [3]
3. A single observation of \(D\) is made. The value obtained, \(d\), is the common difference of an arithmetic sequence. The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral \(Q\) Find the exact probability that the smallest angle of \(Q\) is more than \(50°\) [5]

The discrete random variable \(X\) takes values 1, 2, 3, 4 and 5, and its probability distribution is defined as follows. $$\mathrm{P}(X = x) = \begin{cases} a & x = 1, \\ \frac{1}{2}\mathrm{P}(X = x - 1) & x = 2, 3, 4, 5, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac{16}{31}\). [2]

The discrete probability distribution for \(X\) is given in the table.

1. Find the probability that \(X\) is odd. [1]

Two independent values of \(X\) are chosen, and their sum \(S\) is found.

1. Find the probability that \(S\) is odd. [2]
2. Find the probability that \(S\) is greater than 8, given that \(S\) is odd. [3]

Sheila sometimes needs several attempts to start her car in the morning. She models the number of attempts she needs by the discrete random variable \(Y\) defined as follows. $$\mathrm{P}(Y = y + 1) = \frac{1}{2}\mathrm{P}(Y = y) \quad \text{for all positive integers } y.$$

1. Find P\((Y = 1)\). [2]
2. Give a reason why one of the variables, \(X\) or \(Y\), might be more appropriate as a model for the number of attempts that Sheila needs to start her car. [1]

The probability distribution for the discrete random variable \(W\) is given in the table.

\(w\)	1	2	3	4
\(P(W = w)\)	0.25	0.36	\(x\)	\(x^2\)

1. Show that \(\text{Var}(W) = 0.8571\). [7]
2. Find \(\text{Var}(3W + 6)\). [1]

A random variable \(X\) has probability distribution given by \(P(X = x) = \frac{1}{860}(1 + x)\) for \(x = 1, 2, 3, \ldots, 40\).

1. Find \(P(X > 39)\). [2]
2. Given that \(x\) is even, determine \(P(X < 10)\). [6]

A game is played as follows. A fair six-sided dice is thrown once. If the score obtained is even, the amount of money, in £, that the contestant wins is half the score on the dice, otherwise it is twice the score on the dice.

1. Find the probability distribution of the amount of money won by the contestant. [3]
2. The contestant pays £5 for every time the dice is thrown. Find the standard deviation of the loss made by the contestant in 120 throws of the dice. [5]

\begin{enumerate}[label=(\alph*)] \item In a game show contestants are asked up to five questions in succession to qualify for the next round. An incorrect answer eliminates a contestant from the game show. Let \(X\) denote the number of questions correctly answered by a contestant. The probability distribution of \(X\) is given below.

\(x\)	0	1	2	3	4	5
\(\mathrm{P}(X = x)\)	0.30	0.25	0.20	0.16	0.06	0.03

1. Find the expected number of correctly answered questions and the variance of the distribution. [3]
2. Find the probability that a randomly selected contestant will correctly answer 3 or more questions. [1]
3. Each show had two contestants. Find the probability that both the contestants will correctly answer at least one question. [2]

\item In a promotion, a newspaper included a token in every copy of the newspaper. A proportion, 0.002, are winning tokens and occur randomly. A reader keeps buying copies of the newspaper until he buys one with a winning token and then stops. Let \(Y\) denote the number of copies bought.

1. Explain briefly why this situation may be modelled by a geometric distribution and write down a formula for \(\mathrm{P}(Y = y)\). [2]
2. Find the probability that the reader gets a winning token with the twentieth copy bought. [2]
3. Find the probability that the reader will not have to buy more than three copies in order to get a winning token. [2] \end{enumerate]

The random variable \(X\) is defined as the difference (always positive or zero) between the scores when 2 ordinary dice are rolled.

1. Copy and complete the probability distribution table for \(X\). [2]

   \(x\)	0	1	2	3	4	5
   P(\(X = x\))

2. Find the expectation and variance of \(X\). [5]

A discrete random variable \(X\) has the following probability distribution.

\(x\)	1	2	\(n\)	7
\(\mathrm{P}(X = x)\)	0.4	0.3	\(p\)	0.1

1. Write down the value of \(p\). [1]
2. Given that \(\mathrm{E}(X) = 2.5\), find \(n\). [2]
3. Find \(\mathrm{Var}(X)\). [3]

A biased tetrahedral die has faces numbered \(1\) to \(4\). The random variable \(X\) is the number on the face of the die which is in contact with the table after the die has been thrown. It is known, for this die, that \(\text{P}(X = x) = kx\) where \(k\) is a constant.

1. Determine the value of \(k\) and state the moment generating function of \(X\). [3]
2. Hence find \(\text{E}(X)\) and \(\text{Var}(X)\). [7]