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
$P (X = x)$

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1b1 mark

Find the probability that John rolls at least one six.

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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 $p$ .

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4b4 marks

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

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

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6b2 marks

Find

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

(ii) $P (X > 2)$

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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) $P (X > 0)$

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7b2 marks

Given that $X$ only takes integer values, complete the following probability distribution table for $X$ .

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

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33 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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5a2 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 $k$ .

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5b2 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
$P (Y = y)$

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7

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 $q$ .

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5a

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 $p$ .

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5b

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 $Y$ .

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6a

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
$P (X = x)$

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6b

2 marks

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

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75 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
$P (X = x)$

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$x$	$a$	$b$	$c$
$P (X = x)$	$\log_{36} a$	$\log_{36} b$	$\log_{36} c$

$d$	10	20	30	40	50
$P (D = d)$	$\frac{k}{10}$	$\frac{k}{20}$	$\frac{k}{30}$	$\frac{k}{40}$	$\frac{k}{50}$

$x$	20	50	80	100
$P (X = x)$	$a$	$b$	$c$	$d$

$r$	2	3
$P (R = r)$	$\frac{1}{4}$	$\frac{3}{4}$

Probability Distributions (Edexcel 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)

Large Data Set

Large Data Set

$g$	1	4
$P (G = g)$	$\frac{2}{3}$	$\frac{1}{3}$