Probability Distributions | DP IB Maths: AI SL Topic Questions 2021

1a

2 marks

Frank plays a game involving a biased six-sided die.

The faces of the die are numbered 1 to 6.

The score of the game,X , is the number which lands face up after the die is rolled.

The following table shows the probability distribution for X.

Score, x	1	2	3	4	5	6
$P (X = x)$	$\frac{1}{6}$	$\frac{1}{2} p$	$\frac{1}{8}$	$\frac{3}{2} p$	$\frac{1}{12}$	3p

Calculate the exact value of p.

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1b

3 marks

Frank plays the game once.

Calculate the expected score.

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1c

3 marks

Frank plays the game twice and adds the scores together.

Find the probability Frank has a total score of 4, giving your answer as a fraction.

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1a

1 mark

Three biased coins are tossed.

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

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1b

3 marks

For each coin the probability of getting heads is $\frac{2}{3}$ . A random variable, X, is defined as the number of heads when the three coins are tossed.

Complete the following probability distribution table for X:

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

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1c

2 marks

Hence, by inserting the relevant probabilities, represent the probability distribution for X as a

piecewise function in the form

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

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1d

2 marks

Represent the probability distribution for $X$ as a bar chart.

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1a

2 marks

Ben plays a game involving a biased eight-sided die.

The faces of the die are labelled -4, -2, -1, 0, 1, 3, 5, 6.

The score of the game, X, is the number which lands face up after the die is rolled.

The following table shows the probability distribution for X.

Score, x	-4	-2	-1	0	1	3	5	6
$P (X = x)$	$\frac{1}{6}$	p	$\frac{1}{8}$	$\frac{1}{4}$	$\frac{1}{12}$	$\frac{1}{8}$	$\frac{1}{48}$	$q$

It is given that $p = 4 q$ .

Calculate the exact values of p and q.

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1b

3 marks

Ben plays the game once.

Calculate the expected score.

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1c

3 marks

Ben plays the game twice and adds the scores together.

Find the probability that Ben has a total score of -3, giving your answer as a fraction in its simplest form.

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Prize, s	0	20	Grand prize
$P (S = s)$	9p	7p	4p

Region	A	B	C	Missed target
Probability	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{a}{15}$	$\frac{b}{15}$

Region	A	B	C	Missed target
Prize	$35	$k	$7.50	0

Region	A	B	C	D	E	Miss
Score	1	4	4	8	8	-2
$P (X = x)$	0.3	p	p	q	q	0.4

Section	A	B	C	D	E	F
Area	$\frac{2}{7}$	$\frac{5}{21}$	$\frac{4}{21}$	$\frac{1}{7}$	$p$	$\frac{1}{21}$
Prize	$6	$5	$4	$3	$2	$1

Number on sector	1	2	3	4	5	6
probability	1.5p	p	$\frac{3}{25}$	$\frac{1}{10}$	$\frac{3}{50}$	$\frac{1}{100}$

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

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

Prize, $s	0	2	10	20	100	Grand prize
$P (S = s)$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{6}$	$\frac{1}{24}$	p	$\frac{1}{1000}$

x	-5	-3	-1	0	1	3	5
$P (X = x)$	0.42	$\frac{k^{4}}{2}$	0.05	0.21	${(\frac{3 k}{4})}^{2}$	0.09	0.07

x	0	1	2	3	4	5	6	7	8
$P (X = x)$	0.32	0.22	0.21	$\frac{1}{k}$	$\frac{7}{k^{2}}$	$\frac{4}{k^{2}}$	$\frac{2}{k^{2}}$	$\frac{1}{k^{2}}$	$\frac{1}{k^{2}}$

Number of apple products, x	$P (X = x)$
0	0.1925
1	0.1815
2	0.2250
3	p
4	0.0895
5	0.0504
6	0.0307
7	0.0104
8	q

x	-5	-3	-1	0	1	3	5
$P (X = x)$	0.24	2k²	0.04	0.12	3k	0.19	0.11

x	-3	-1	0	1	3
$P (X = x)$	0.11	k²	0.1	2k	0.1

Region	A	B	C	D	E
Probability	0.55	0.15	0.15	0.1	0.05

Region	A	B	C	D	E
Points	-5	2	3	10	k

DP IB Maths: AI SL

Topic Questions

4.4 Probability Distributions