Probability Distributions

1a2 marks

John has two fair six-sided dice. Each one is labelled with the numbers 1 to 6.

The discrete random variable, $X$ , is defined as the number of sixes obtained when John rolls the two dice once.

Complete the following probability distribution table for $X$ .

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

How did you do?

1b1 mark

Find the probability that John rolls at least one six.

How did you do?

Did this page help you?

4a2 marks

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

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

Find the value of $p$ .

How did you do?

4b4 marks

Find $P (3 \leq X \leq 7)$

How did you do?

Did this page help you?

6a1 mark

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

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

Complete the following cumulative probability function table for $X$

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

How did you do?

6b2 marks

Find

(i) $P (X \leq 3)$

(ii) $P (X > 2)$

How did you do?

Did this page help you?

7a2 marks

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

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

Find:

(i) $P (X < 0)$

(ii) $P (X > 0)$

How did you do?

7b2 marks

Given that $X$ only takes integer values, 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?

Did this page help you?

13 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
$P (X = x)$

How did you do?

Did this page help you?

3a2 marks

A spinner has four sections labelled 1, 3, 5 and 7. The spinner is spun and the number it lands on is represented by the random variable $X$ which 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?

3b2 marks

The spinner is spun twice. The random variable $Y$ represents the number of times that the spinner lands on the section labelled 3.

Complete the probability distribution of $Y$ .

$y$	0	1	2
$P (Y = y)$

How did you do?

Did this page help you?

5

4 marks

A discrete random variable $X$ has the probability distribution

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

Given that $P (X = 0) = P (X > 2)$ , find the value of $p$ and the value of $q$ .

How did you do?

Did this page help you?

3a

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?

3b

5 marks

The random variables $X_{1}$ and $X_{2}$ are independent and each have the same distribution as $X$ .

The random variable $Y$ is defined as $Y = X_{1} \times X_{2}$ , the product of $X_{1}$ and $X_{2}$ .

Fully describe the probability distribution for $Y$ .

How did you do?

Did this page help you?

4a

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 it lands on a 6 or he has rolled the dice four times. The random variable $X$ is defined as the number of times that the dice is rolled.

Complete the probability distribution of $X$ .

$x$	1	2	3	4
$P (X = x)$

How did you do?

4b

2 marks

Find $P (X^{2} \geq 5 X - 5)$ .

How did you do?

Did this page help you?

55 marks

Two biased coins are tossed. For each coin, the probability of getting heads is $\frac{1}{3}$ . The number of heads is represented by the random variable $H$ .

A fair spinner with three sectors numbered 1 to 3 is spun. The number it lands on is represented by the random variable $S$ .

The random variable, $X$ , is defined as the product of the number of heads and the number on the spinner, such that $X = H \times S$ .

Complete the following probability distribution for $X$ .

$x$	0	1	2	3	4	6
$P (X = x)$

How did you do?

Did this page help you?

Probability Distributions (AQA A Level Maths: Statistics): Exam Questions

Statistical Sampling

Sampling & Data Collection

Data Presentation & Interpretation

Statistical Measures

Data Presentation

Working with Data

Correlation & Regression

Further Correlation & Regression

Probability

Basic Probability

Further Probability

Statistical Distributions

Binomial Distribution

Normal Distribution

Working with Distributions

Hypothesis Testing

Introduction to Hypothesis Testing

Hypothesis Testing (Binomial Distribution)

Hypothesis Testing (Normal Distribution)