3.1 Discrete Random Variables

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3 marks

The discrete random variable, $X$ , is defined as the number of sixes obtained from rolling two fair dice.

(i)

Find the probability of obtaining two sixes from rolling two fair dice.

(ii)

Complete the following probability distribution table for

X

$x$	0	1	2
$P (X = x)$	$\frac{25}{36}$

How did you do?

2 marks

Use the table, or otherwise, to find the probability of obtaining at least one six from rolling two fair dice.

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1 mark

The discrete random variable $X$ has the probability function

$P (X = x) \{\begin{cases} \frac{1}{4} & x = 0, 1, 2, 3 \\ 0 & otherwise \end{cases}$

Briefly explain why $X$ has a uniform probability distribution.

How did you do?

2 marks

Find:

(i)

P (1 \leq X \leq 2)

(ii)

P (X < 3)

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2 marks

The discrete random variable $X$ has the probability function

$P (X = x) = \{\begin{cases} k x & x = 2, 3 \\ 0 & otherwise \end{cases}$

Use the fact that the sum of all probabilities equals 1 to show that $k = 0.2$ .

How did you do?

2 marks

Write down:

(i)

P (2 \leq X < 3)

(ii)

P (X = 5)

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2 marks

A discrete random variable has the probability distribution shown in the following table:

$x$	2	4	6	8	10
$P (X = x)$	$\frac{2}{5}$	$\frac{1}{10}$	$\frac{1}{5}$	$p$	$\frac{1}{10}$

Use the fact that the sum of all probabilities equals 1 to find the value of $p$ .

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4 marks

Find:

(i)

P (X \leq 4)

(ii)

P (X > 7)

(iii)

P (2 \leq X \leq 6)

(iv)

P (3 < X < 7)

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2 marks

The discrete random variable $X$ has the probability function

$P (X = x) = \{\begin{matrix} k x & x = 1, 3 \\ \frac{k x}{2} & x = 2, 4 \\ 0 & otherwise \end{matrix}$

Use the fact that the sum of all probabilities equals 1 to show that $k = \frac{1}{7}$ .

How did you do?

1 mark

Briefly explain why $X$ has a non-uniform probability distribution.

How did you do?

2 marks

Show that $P (X \leq 2) = P (X = 4)$ .

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2 marks

The discrete random variable has the probability distribution shown in the following table:

$x$	1	2	3	4	5
$P (X = x)$	$\frac{5}{12}$	$\frac{2}{12}$	$\frac{1}{12}$	$\frac{3}{12}$	$\frac{1}{12}$

Complete the following cumulative probability function table for :

$x$	1	2	3	4	5
$P (X \leq x)$	$\frac{5}{12}$	$\frac{7}{12}$			$1$

How did you do?

5 marks

Use your table from part (a) to find:

(i)

F (3)

(ii)

P (X \geq 4)

(iii)

P (2 \leq X \leq 4)

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3 marks

The discrete random variable $X$ has the cumulative probability distribution shown in the following table:

$x$	-2	-1	0	1	2
$P (X \leq x)$	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{3}{5}$	$\frac{4}{5}$	$\frac{5}{5}$

Complete the following probability distribution table for $X$ :

$x$	-2	-1	0	1	2
$P (X = x)$	$\frac{1}{5}$	$\frac{1}{5}$

How did you do?

2 marks

Find:

(i)

P (X < 0)

(ii)

P (X > 0)

How did you do?

2 marks

Explain, with a reason, whether $X$ has a uniform probability distribution or not.

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2 marks

The discrete random variable $X$ has the probability distribution shown in the following table:

$x$	1	2	3	4	5
$P (X = x)$	$\frac{5}{12}$	$\frac{2}{12}$	$\frac{1}{12}$	$\frac{3}{12}$	$\frac{1}{12}$

Use the formula $E (X) = \sum x p$ to show that $E (X) = \frac{29}{12}$ .

How did you do?

2 marks

Use the formula $E (X^{2}) = \sum x^{2} p$ to show that $E (X^{2}) = \frac{95}{12}$ .

How did you do?

1 mark

Write down the formula that links $Var (X), E (X)$ and $E (X^{2})$ .

How did you do?

1 mark

Hence show that $Var (X) = \frac{299}{144}$ .

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1 mark

The discrete random variable $X$ has the probability function

$P (X = x) = \{\begin{matrix} \frac{1}{4} & x = 0 \\ \frac{1}{8} & x = 1, 2 \\ \frac{5}{16} & x = 3 \\ p & x = 4 \\ 0 & otherwise \end{matrix}$

Briefly explain how you can deduce that $p = \frac{3}{16}$ .

How did you do?

2 marks

Find $E (X) .$

How did you do?

2 marks

Show that $E (X^{2}) = \frac{103}{16}$ .

How did you do?

2 marks

Hence find $Var (X)$ .

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10a

1 mark

The discrete random variable $X$ has the probability distribution shown in the following table:

$x$	1	2	3	4	5
$P (X = x)$	$\frac{1}{5}$	$\frac{1}{10}$	$p$	$p$	$q$

Use the fact that the sum of all probabilities equals 1 to show that $2 p + q = \frac{7}{10}$ .

How did you do?

10b

2 marks

Given that $E (X) = \frac{33}{10}$ , use the formula $E (X) = \sum x p$ to show that $7 p + 5 q = \frac{29}{10}$ .

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10c

2 marks

Hence simultaneously solve the equations in part (a) and part (b) to find the values of $p$ and $q$ .

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6 marks

$X$ is a random variable such that $E (X) = 5$ and $Var (X) = 3$ .

Using the formulae $E (a X + b) = a E (X) + b$ and $Var (a X + b) = a^{2} Var (X)$ , find the mean and variance of the following random variables:

(i)

4 X + 1

(ii)

7 X - 2

(iii)

5 - X

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1 mark

Three biased coins are tossed.

Write down all the possible outcomes when the three coins are tossed.

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3 marks

A random variable, $X$ , is defined as the number of heads when the three coins are tossed minus the number of tails.

Given that for each coin the probability of getting heads is $\frac{3}{5}$ ,

complete the following probability distribution table for $X$ :

$x$
$P (X = x)$

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4 marks

The random variable $X$ has the probability function

$P (X = x) = \{\begin{cases} \frac{1}{k} & x = 1, 2, 3, 4, 5, 6 \\ 0 & otherwise . \end{cases}$

(i)

Write down the value of

k

(ii)

Write down the name of this probability distribution.

(iii)

Find the values of

E (X)

and

Var (X)

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2 marks

A student claims that a random variable $X$ has a probability distribution defined by the following probability mass function:

$P (X = x) = \{\begin{cases} \frac{x^{2}}{30} & x = - 1, 1, 3, 5 \\ 0 & otherwise \end{cases}$

Explain how you know that the student’s function does not describe a probability distribution

How did you do?

2 marks

Given that the correct probability mass function is of the form

$P (X = x) = \{\begin{cases} \frac{x^{2}}{k} & x = - 1, 1, 3, 5 \\ 0 & otherwise \end{cases}$

where $k$ is a constant,

find the exact value of $k$ .

How did you do?

2 marks

Find $P (X > 0)$ .

How did you do?

1 mark

State, with a reason, whether or not $X$ is a discrete random variable.

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2 marks

The random variable $X$ has the probability function

$P (X = x) = \{\begin{matrix} 0.21 & x = 0, 1 \\ k x & x = 3, 6 \\ 0.11 & x = 10, 15 \\ 0 & otherwise \end{matrix}$

Find the value of $k$ .

How did you do?

2 marks

Construct a table giving the probability distribution of $X .$

How did you do?

1 mark

Find $P (3 < X \leq 14)$ .

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1 mark

A discrete random variable $X$ has the probability distribution shown in the following table:

$x$	$- 1$	$1$	$2$
$P (X = x)$	$\frac{5}{12}$	$p$	$\frac{1}{4}$

Find the value of $p$ .

How did you do?

5 marks

$X$ is sampled twice such that the results of the two experiments are independent of each other, and the outcomes of the two experiments are recorded. A new random variable, $Y$ , is defined as the sum of the two outcomes.

Complete the following probability distribution table for $Y$ :

$y$	$- 2$	$0$	$1$	$2$	$3$	$4$
$P (Y = y)$

How did you do?

4 marks

Find:

(i)

P (Y \neq 0)

(ii)

P (Y > 1)

(iii)

P (- 2 < Y < 2)

(iv)

P (Y < 0 or Y \geq 2)

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4 marks

Leonidas is playing a game with a fair six-sided dice on which the faces are numbered 1 to 6. He rolls the dice until either a ‘6’ appears or he has rolled the dice four times. The random variable $X$ is defined as the number of times that the dice is rolled.

Write down the probability distribution of $X$ in table form.

How did you do?

2 marks

Complete the following cumulative probability function table for $X$ :

$x$	1	2	3	4
$P (X \leq x)$

How did you do?

2 marks

Find the probability that $X$ is

(i)

at most 3

(ii)

at least 3.

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1 mark

The discrete random variable $X$ has the probability distribution shown in the following table:

$x$	$- 2$	$0$	$2$	$4$	$6$
$P (X = x)$	$p$	$\frac{2}{15}$	$\frac{1}{4}$	$\frac{2}{15}$	$p$

Without working out the value of $p$ , explain why $E (X) = 2$ .

How did you do?

1 mark

Find the value of $p .$

How did you do?

2 marks

Find $Var (X) .$

How did you do?

2 marks

The outcome of a random variable $Y$ is double the outcome of $X$ .

Complete the probability distribution for $Y$ below:

$y$	$- 4$	$0$	$4$	$8$
$P (Y = y)$		$\frac{2}{15}$

How did you do?

2 marks

Find $P (Y > X)$ .

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5 marks

The discrete random variable $X$ has the probability distribution shown in the following table:

$x$	$0$	$1$	$2$	$3$	$4$
$P (X = x)$	$p$	$p$	$0.2$	$0.1$	$q$

It is given that $E (X) = 2.45 .$

Find the values of $p$ and $q$ .

How did you do?

2 marks

Find $Var (X) .$

How did you do?

2 marks

Find $P (X < E (X)) .$

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6 marks

A discrete random variable, $X$ , can take the values 1, 3, 5 or 7 and it has the cumulative distribution function $F (x)$ given in the table.

$x$	1	3	5	7
$F (x)$	0.5	0.65	0.9	1

Find the value of:

(i)

P (X = 3)

(ii)

P (3 X + 3 > X + 9)

(iii)

E (X)

(iv)

Var (X)

How did you do?

5 marks

Find the value of:

(i)

E (\frac{X}{2} + 7)

(ii)

Var (\frac{2 - x}{5})

(iii)

E (3 X^{2} - 1)

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1 mark

Three biased coins are tossed.

Write down all the possible outcomes when the three coins are tossed.

How did you do?

3 marks

A random variable, $X$ , is defined as the number of heads when the three coins are tossed.

Given that for each coin the probability of getting heads is $\frac{2}{3}$ ,

complete the following probability distribution table for $X$ :

$x$	0	1	2	3
$P (X = x)$

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3 marks

The random variable $X$ has the probability function

$P (X = x) = \{\begin{cases} \frac{1}{k} & x = 1, 2, 3, 4, 5 \\ 0 & otherwise \end{cases}$

(i)

Show that

k = 5

(ii)

Write down the name of this probability distribution.

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2 marks

The random variable $X$ has the probability function

$P (X = x) = \{\begin{cases} k x & x = 1, 3, 5, 7 \\ 0 & otherwise \end{cases}$

Find the value of $k$ .

How did you do?

2 marks

Find $P (X > 3)$ .

How did you do?

1 mark

State, with a reason, whether or not $X$ is a discrete random variable.

How did you do?

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2 marks

The random variable $X$ has the probability function

$P (X = x) = \{\begin{matrix} 0.23 & x = - 1, 4 \\ k & x = 0, 2 \\ 0.13 & x = 1, 3 \\ 0 & otherwise \end{matrix}$

Find the value of $k$ .

How did you do?

2 marks

Construct a table giving the probability distribution of $X$ .

How did you do?

1 mark

Find $P (0 \leq X < 3)$ .

How did you do?

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5 marks

A discrete random variable $X$ has the probability distribution shown in the following table:

$x$	0	1	2	3	4
$P (X = x)$	$\frac{5}{24}$	$\frac{1}{3}$	$\frac{1}{4}$	$\frac{1}{12}$	$\frac{1}{8}$

Find:

(i)

P (X < 4)

(ii)

F (2.5)

(iii)

P (X > 1)

(iv)

P (2 < X \leq 4)

(v)

P (0 < X < 4)

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1 mark

Leonardo has constructed a biased spinner with six sectors labelled $0, 1, 1, 2, 3$ and $5$ . The probability of the spinner landing on each of the six sectors is shown in the following table:

number on sector	0	1	1	2	3	5
probability	$\frac{6}{20}$	$p$	$\frac{3}{20}$	$\frac{5}{20}$	$\frac{3}{20}$	$\frac{1}{20}$

Find the value of $p$ .

How did you do?

3 marks

Leonardo is playing a game with his biased spinner. The score for the game, $X$ , is the number which the spinner lands on after being spun.

Leonardo plays the game twice and adds the two scores together. Find the probability that Leonardo has a total score of 5.

How did you do?

2 marks

Complete the following cumulative probability function table for $X$ :

Score $x$	0	1	2	3	5
$P (X \leq x)$	$\frac{6}{20}$				$1$

How did you do?

2 marks

Find the probability that $X$ is

(i)

no more than 1

(ii)

at least 3.

How did you do?

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1 mark

The discrete random variable $X$ has the probability distribution shown in the following table:

$x$	2	3	5	7	11
$P (X = x)$	$\frac{1}{4}$	$\frac{1}{3}$	$p$	$\frac{1}{6}$	$\frac{1}{12}$

Find the value of $p$ .

How did you do?

2 marks

Find $E (X)$ .

How did you do?

2 marks

Find $Var (X)$ .

How did you do?

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2 marks

The discrete random variable $X$ has the probability distribution shown in the following table:

$x$	$- 2$	$0$	$2$	$4$	$6$
$P (X = x)$	$p$	$\frac{1}{2}$	$q$	$\frac{1}{15}$	$q$

It is given that $E (X) = 0$ .

Show that $p - 4 q = \frac{2}{15}$ .

How did you do?

1 mark

Write down a second equation involving $p$ and $q$ .

How did you do?

2 marks

Hence find the values of $p$ and $q$ .

How did you do?

2 marks

Find $Var (X)$ .

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4 marks

The random variable $X$ has mean $μ$ and variance $σ^{2}$ . Given that

$E (2 X + 7) = 16 Var (2 X + 7) = 5,$

find the value of $μ$ and the value of $σ^{2}$ .

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1 mark

Two biased coins are tossed and a fair spinner with three sectors numbered 1 to 3 is spun.

Write down all the possible outcomes when the two coins are tossed and the spinner is spun.

How did you do?

5 marks

A random variable, $X$ , is defined as the number of heads when the two coins are tossed multiplied by the number the spinner lands on when it is spun.

For each coin the probability of getting heads is $\frac{1}{3}$ .

Complete the following probability distribution table for $X$ :

$x$	0	1	2	3
$P (X = x)$

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2 marks

A student claims that a random variable $X$ has a probability distribution defined by the following probability mass function:

$P (X = x) = \{\begin{matrix} \frac{1}{3 x^{2}} & x = - 3, - 1 \\ \frac{1}{3 x^{3}} & x = 1, 3 \\ 0 & otherwise \end{matrix}$

Explain how you know that the student’s function does not describe a probability distribution.

How did you do?

2 marks

Given that the correct probability mass function is of the form

$P (X = x) = \{\begin{matrix} \frac{k}{x^{2}} & x = - 3, - 1 \\ \frac{k}{x^{3}} & x = 1, 3 \\ 0 & otherwise \end{matrix}$

where $k$ is a constant,

Find the exact value of $k$ .

How did you do?

2 marks

Find $P (X < 2)$ .

How did you do?

1 mark

State, with a reason, whether or not $X$ is a discrete random variable.

How did you do?

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4 marks

The random variable $X$ has the probability function

$P (X = x) = \frac{x^{2}}{495}, x = p, 2 p, 3 p, 4 p, 5 p$

where $p > 0$ is a constant.

Construct a table giving the probability distribution of $X$ .

How did you do?

4 marks

Find:

(i)

the mean,

μ

(ii)

the standard deviation,

σ

of $X$ .

How did you do?

2 marks

Find $P (μ - σ < X < μ + σ)$ .

How did you do?

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6 marks

The independent random variables $X$ and $Y$ have probability distributions

$P (X = x) = p, x = 1, 2, 3, 5, 8, 11 P (Y = y) = \frac{q}{y}, y = 1, 3, 6$

where $p$ and $q$ are constants.

Find $P (X > Y)$ .

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6 marks

Leofranc is playing a gambling game with a fair six-sided dice on which the faces are numbered 1 to 6. He must pay £2 to play the game. He then chooses a ‘lucky number’ between 1 and 6, and rolls the dice until either his lucky number appears or he has rolled the dice four times. If his lucky number appears on the first roll, he receives £5 back. If his lucky number appears on the second, third or fourth rolls, he receives £3, £2 or £1 back respectively. If his lucky number has not appeared by the fourth roll, then the game is over and he receives nothing back.

The random variable $W$ is defined to be Leofranc’s profit (i.e., the amount of money he receives back minus the cost of playing the game) when he plays the game one time. Note that a negative profit indicates that Leofranc has lost money on the game.

Draw up the probability distribution table for $W$ .

How did you do?

3 marks

Find the probability that when playing the game one time Leofranc

(i)

wins money

(ii)

loses money

(iii)

breaks even (i.e. does not win or lose money).

How did you do?

2 marks

Find the expected profit after one game.

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1 mark

The discrete random variable $X$ has the probability distribution shown in the following table:

$x$	$- 3$	$- 2$	-1	$0$	$1$
$P (X = x)$	$p$	$q$	$0.1$	$q$	$p$

Write down the value of $E (X)$ .

How did you do?

1 mark

It is given that $E (X^{2}) = 3.4$ .

Find $Var (X)$ .

How did you do?

5 marks

Find the values for $p$ and $q$ .

How did you do?

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1 mark

A spinner has three sectors labelled 0, 1 and 2. Let $X$ be the random variable denoting the number the spinner lands on when spun. The probability distribution table for $X$ is shown below:

$x$	$0$	$1$	2
$P (X = x)$	$a$	$b$	$c$

It is given that $E (X) = 1.1$ and $Var (X) = 0.89$ .

Write down the value of $E (X^{2})$ .

How did you do?

4 marks

Find the values of $a, b$ and $c$ .

How did you do?

4 marks

Susie spins the spinner twice and adds together the two numbers to calculate her score, $S$ . Tommy spins the spinner once and doubles the number to calculate his score, $T$ . Each spin of the spinner is independent of all other spins.

Draw up the probability distribution table for:

(i)

S,

(ii)

T

How did you do?

1 mark

Which player is most likely to get a score that is bigger than 2?

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7 marks

The probability of rolling a 6 on a biased die is $p$ . In a game, this die is rolled twice and then the player multiplies the number of times the die lands on a 6 by 50 and then adds 5 to calculate the score.

(i)

Given that the mean score is 15, draw up a probability distribution table for the number of times the die lands on a 6 when rolled twice.

(ii)

Find the variance of the scores.

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4 marks

All the odd numbers from 1 to 99 inclusive are written on individual pieces of paper and placed inside a bag. A piece of paper is picked at random and the odd number $Y$ is recorded.

Find the expectation and standard deviation of $Y$ .

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10a

5 marks

A biased four-sided dice is rolled and the number that it lands on is denoted which has probability distribution shown below.

$x$	$- 2$	$- 1$	$\frac{1}{2}$	$2$
$P (X = x)$	$a$	$b$	$c$	$c$

The cumulative distribution function of $X$ is given by:

$x$	$- 2$	$- 1$	$\frac{1}{2}$	$2$
$F (x)$	$\frac{1}{7}$	$\frac{2}{5}$	$d$	$e$

Find the values of $a, b, c, d,$ and $e$ .

How did you do?

10b

2 marks

Saskia and Tamara place a game by rolling the dice. Saskia’s score is the number the dice lands on and Tamara’s score is the reciprocal of that number.

Find the probability that Saskia’s score is bigger than Tamara’s score.

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Edexcel International AS Maths: Statistics 1

Topic Questions

3.1 Discrete Random Variables