HIGHER SCHOOL CERTIFICATE · conditional probability and the formula for conditional ... The...

HIGHERSCHOOLCERTIFICATEYEAR11ADVANCEDMATHEMATICS

PROBABILITYANDSTATISTICS

FREESAMPLE

1

First Published by John Kinny-Lewis in 2018 John Kinny-Lewis 2018 National Library of Australia Cataloguing-in-publication data ISBN: 978-0-6484118-9-5 This book is copyright. Apart from any fair dealing for purposes of private study, research, criticism or review as permitted under the Copyright Act 1968, no part may be reproduced, stored in a retrieval system, or transmitted, in any form by any means, electronic, mechanical, photocopying, recording, or otherwise without prior written permission. Enquiries to be made to John Kinny-Lewis. Copying for educational purposes. Where copies of part or the whole of the book are made under Section 53B or Section 53D of the Copyright Act 1968, the law requires that records of such copying be kept. In such cases the copyright owner is entitled to claim payment. Typeset by John Kinny-Lewis Edited by John Kinny-Lewis

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CONTENTS SET 1 5 SET 1 ANSWERS 17 SET 1 SOLUTIONS 18 SET 2 30 SET 2 ANSWERS 42 SET 2 SOLUTIONS 43 SET 3 55 SET 3 ANSWERS 67 SET 3 SOLUTIONS 68 SET 4 80 SET 4 ANSWERS 92 SET 4 SOLUTIONS 93

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PREFACE This book provides a thorough revision of probability and statistics for the new Year 11 Advanced Mathematics HSC syllabus. The book is divided into four sets. Each set contains six topic areas.

• Probability review provides questions on: sample spaces, the probability of an event, experimental probability.

• The complement, union and intersection provides questions on: complementary events, Venn diagrams involving union and intersection, two-way tables, the addition rule P(A ∪B) = P(A)+ P(B)− P(A ∩B).

• Conditional probability provides questions on: the notion of

conditional probability and the formula for conditional

probability P(A/B) = P(A ∩B)

P(B).

• Independent events provides questions on: the notion of

independence and the formulas for independence, P(A/B) = P(A) , P(A)× P(B) = P(A ∩B) .

• Sampling with and without replacement provides questions on: arrays and tree diagrams to determine outcomes, probabilities for multi-stage problems.

• Discrete probability distributions provides questions on: non-uniform discrete random variables, the expected value, E(X) = Σx

ip

i, the mean µ = E(X) , the variance Var (X ) = Σx

i2p

i− µ2

Each topic area contains six questions with space for the student’s working. Answers and detailed solutions are provided at the end of each set. Harder questions are marked with #.

4

SET 1 Probability Review Question 1

From the english alphabet, a letter is selected at random.What is the chance of drawing(i) a vowel(ii) a consonant(iii) one of the letters of the word 'focus'

Question 2

List the six different orders in which Holly, Polly and Molly may sit in a row.If the three of them sit randomly in a row, determine the probability that:(i) Holly sits in the middle.(ii) Polly and Molly are seated together.

Question 3

A bag contains 12 balls. Three of these are black and nine are red.A ball is taken from the bag at random. What is the probability that it is red?

5

SET 1 Probability Review Question 4

Fifty-five balls, numbered 1 to 55, are placed in a bag, and one ball is drawnat random. What is the probability that the number drawn is even? Question 5

Three coins are tossed simultaneously.(a) List the possible outcomes.(b) Find the probability of getting:

(i) three heads(ii) at least one head.

Question 6

An experiment involves tossing three coins and counting the number of heads.Below are the results after running the experiment 100 times.

Number of heads 0 1 2 3 Frequency 8 39 46 7

Find the experimental probability of obtaining:(i) zero heads(ii) fewer than two heads.

6

SET 1 The Complement, Union and Intersection Question 1

A number is chosen at random from the first 24 positive whole numbers.What is the probability that it is not divisible by 4 ?

Question 2

Let A be the set of all factors of 6 and B the set of all positive even integersless than 12. Find:(i) A ∪B(ii) A ∩B

Question 3

The Venn diagram shows thedistribution of elements in twosets A and B. Transfer the information in the Venn diagramto a two-way table and find:(i) P A ∩B( )(ii) P A ∩B'( )

7

SET 1

The Complement, Union and Intersection

Question 4

From a group of 40 adults, 24 enjoy reading crime fiction (C), 14 enjoyreading historical fiction (H) and 6 enjoy reading crime fiction andhistorical fiction.(a) Illustrate the information on a Venn diagram.(b) Find the probability that a person chosen at random from the group will enjoy reading: (i) crime fiction (ii) only crime fiction

Question 5

In a group of 26 students, 18 play tennis (T) and 12 play hockey (H). Assuming that each of the 26 students plays at least one of these sports, find the probability that a student chosen randomly from this group plays both tennis and hockey.

Question 6

Use the Venn diagram to find:(a) (i) P(A) (ii) P(B)

(iii) P(A ∩B) (iv) P(A ∪B)(b) Hence show that:

P(A ∪B) = P(A)+ P(B)− P(A ∩B)

8

SET 1

Conditional Probability

Question 1

Find P(A/B) if P(A ∩B) = 0.1 and P(B) = 0.5. Question 2

A pair of dice are rolled. Given that the sum of the rolls is 6, find theprobability that at least one of dice rolls a 2.

Question 3

Use the Venn diagram to find:(i) n(A ∩B)(ii) n(B)

(iii) Using P(A/B) = n(A ∩B)n(B)

,find P(A/B)

9

SET 1

Conditional Probability

Question 4

Use the two way table to find:(i) n(A ∩B)(ii) n(A)(iii) P(B/A)

A A’ B 2 6 8 B’ 3 5 8

5 11 16

Question 5

Two events, A and B are mutually exclusive. What is the value of:(i) P(A/B)(ii) P(B/A)

Question 6

Two events A and B, are such that B is a subset of A, as shown in theadjacent diagram. Find:(i) P(A/B) (ii) P(B/A)

10

SET 1

Independent Events

Question 1

The probability that a person shops at Molesworth (M) is 34

and the

probability that the person is female (F) is 12

. If these events are

independent, find the probability that the customer of Molesworth is female.

Question 2

Given that P(A) = 14

and P(B) = 13

and P(A ∪B) = x.

Find x if :(i) A and B are mutually exclusive. (ii) A and B are independent.

Question 3

Consider the events A and B, with the number of elements contained in each event given in the adjacent Venn diagram.Given that the events A and B areindependent, find the value of x.

11

SET 1

Independent Events

Question 4

For two independent events A and B, P(A ∪B) = 0.8 and P(B) = 0.4. Find P(A). Question 5

For the events A and B, with details provided in the adjacent two-waytable, complete the table and decideif the events A and B are independent.

A A’ B 1 3 B’ 4 12

Question 6

For the independent events A and B and given the variables provided in the Venn diagram, show that ac = bd.

12

SET 1

Sampling With and Without Replacement

Question 1

A die is tossed three times. What is the probability of obtaining:(i) Three sixes ?(ii) Exactly two sixes ?(ii) At least one six ?

Question 2

A bag contains 6 red balls and 4 black balls. A ball is taken and its colour noted. It is not replaced. A second ball is taken and its colour noted. Find the probability of obtaining: (i) Two red balls. (ii) Two balls of a different colour.

Question 3

A factory assembles torches. Each torch has one battery (Ba) and one bulb (B

u).

It is known that 5% of all batteries and 3% of all bulbs are defective.Find the probability that, in a torch selected at random, both the battery andthe bulb are not defective.

13

SET 1

Sampling With and Without Replacement

Question 4

A fair 4-sided die is rolled twice.(a) List the sample space in the adjacent

table.(b) Find the probability of:

(i) a double (ii) a sum of at least 5 (iii) a sum not equal to 3

1 2 3 4 1 2 3 4

Question 5

A group of 20 children met at MacDougalls, 13 had a Big Doug,15 had a thick shake and9 had a Big Doug and a thick shake. Complete the two-way table provided and find the probability that:(i) a child only had a Big Doug.(ii) a child had a Big Doug given that they had a thick shake.

BD BD’ TS 9 15 TS’

13 20

Question 6 Tom (T) and Jerry (J) are playing a game of Chess. They will play two games and each has an equal chance of winning the first game. If Tom wins the first game, his probability of winning the second game is increased to 0.7. If Tom loses the first game, his probability of winning is reduced to 0.4. (i) Complete the adjacent tree diagram. (ii) Find the probability that Tom wins exactly one game.

14

SET 1

Discrete Probability Distributions

Question 1

A fair coin is thrown twice. The random variable X is the number of headsobtained. Tabulate the probability distribution of X.

x 0 1 2 Total

P X = x( ) 1

Question 2

Two 4-sided dice are thrown simultaneously. The random variable Dis the difference between the smaller and the larger score, or zero if they arethe same. Using the data from the adjacent diagram, tabulate the probability distribution of D.

1 2 3 4 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 4 3 2 1 0

d 0 1 2 3 Total

P D = d( ) 1

Question 3

In the following probability distribution, c is a constant. Find the value of c.

x 0 1 2 3

P X = x( )

13

14

15

c

15

SET 1

Discrete Probability Distributions

Question 4

The probability distribution of the random variable S is given in the following table, where c is a constant. Find the value of c.

s 1 2 3 4 5

P S = s( ) c 3c

c2 c

2 95 128 Question 5

Find the expected value of the variable X, which has the probability distribution given below.

x 1 2 3 4 5 6 Total

P X = x( ) 1/6 1/6 1/6 1/6 1/6 1/6 1

Question 6

Find the expected value and variance of the variable X, which has the probability distribution given below.

x 1 2 3 4 5

P X = x( ) 0.2 0.3 0.2 0.1 0.2

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SET 1 ANSWERS Probability Review Independent Events

1) (i)

526

(ii) 2126

(iii) 526

1)

38

2) (i)

13

(ii) 23

2) (i) x = 7

12 (ii) x = 1

2

3)

34

3) x = 20

4)

2755

4)

23

5)

(a) ξ = HHH,HHT,HTH,THH,

HTT,THT,TTH,TTT⎧⎨⎩

⎫⎬⎭

(b) (i) 18

(ii) 78

5) independent

6) (i)

225

(ii) 47

100 6)

proof

The Complement, Union and Intersection

Sampling With and Without Replacement

1)

34

1) (i)

1216

(ii) 572

(iii) 91

216

2) (i) 1,2,3,4,6,8,10{ } (ii) 2,6{ } 2)

(i)

13

(ii) 8

15

3) (i)

217

(ii) 3

17 3) 0.9215

4) (i)

35

(ii) 920

4) (i)

14

(ii) 58

(iii) 78

5)

213

5) (i)

1320

(ii) 35

6) see solution for proof. 6) 0.35

Conditional Probability Discrete Probability Distributions

1) 0.2 1) see table

2)

25

2) see table

3) (i) 2 (ii) 8 (iii) 0.25 3)

c = 13

60

4) (i) 2 (ii) 5 (iii)

25

4) c = 1

16

5) (i) 0 (ii) 0 5)

3

12

6) (i) 1 (ii) 0.35

6)

E(X) = 2.8, Var(X) =1.96

17

SET 1 SOLUTIONS Probability Review Question 1

From the english alphabet, a letter is selected at random.What is the chance of drawing(i) a vowel(ii) a consonant(iii) one of the letters of the word 'focus'

(i) There are 26 letters in the alphabet and 5 vowels.

∴ P(vowel) = 526

(ii) There are 21 consonants.

∴ P(consonant) = 2126

(iii) There are 5 different letters in the word 'focus'.

∴P(f,o,c,u,s) = 526

Question 2

List the six different orders in which Holly, Polly and Molly may sit in a row.If the three of them sit randomly in a row, determine the probability that:(i) Holly sits in the middle.(ii) Polly and Molly are seated together.

(i) Sample space, ξ = HPM,HMP,PHM,PMH,MPH,MHP{ }Sample space for Holly in the middle, ξ = PHM,MHP{ }∴Probability P H middle( ) = 2

6= 1

3(ii) Sample space for Polly and Molly together, ξ = HPM,HMP,PMH,MPH{ }∴Probability P PM together( ) = 4

6= 2

3

OR Probability P PM together( ) =1− P H middle( ) =1− 13= 2

3

Question 3

A bag contains 12 balls. Three of these are black and nine are red.A ball is taken from the bag at random. What is the probability that it is red?

Number of black balls = 3Number of red balls = 9Total number of balls =12

∴ P Red( ) = 912

= 34

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SET 1 SOLUTIONS Probability Review Question 4

Fifty-five balls, numbered 1 to 55, are placed in a bag, and one ball is drawnat random. What is the probability that the number drawn is even?

552

= 27.5

∴ There are 27 even and 28 odd numbers, since the numbers began withan odd and ended with an odd number.

∴ P(even) = 2755

Question 5

Three coins are tossed simultaneously.(a) List the possible outcomes.(b) Find the probability of getting:

(i) three heads(ii) at least one head.

(a) Sample space is ξ = HHH,HHT,HTH,THH,HTT,THT,TTH,TTT{ }(b) (i) P HHH( ) = 1

8(ii) P at least one H( ) =1− P TTT( )

=1− 18

= 78

Question 6

An experiment involves tossing three coins and counting the number of heads.Below are the results after running the experiment 100 times.

Number of heads 0 1 2 3 Frequency 8 39 46 7

Find the experimental probability of obtaining:(i) zero heads(ii) fewer than two heads.

(i) P zero H( ) = 8100

= 225

(ii) P fewer than 2 H( ) = 8 + 39100

= 47100

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SET 1 SOLUTIONS The Complement, Union and Intersection Question 1

A number is chosen at random from the first 24 positive whole numbers.What is the probability that it is not divisible by 4 ?

Let F = the numbers divisible by 4.∴ the set F = 4,8,12,16,20,24{ }

n(F) = 6 → P(F) = 624

= 14

∴ The probability that the number is NOT divisible by 4 =1− P(F)

= 34

Question 2

Let A be the set of all factors of 6 and B the set of all positive even integersless than 12. Find:(i) A ∪B(ii) A ∩B

A = 1,2,3,6{ } and B = 2,4,6,8,10{ }(i) A ∪B = 1,2,3,4,6,8,10{ }(ii) A ∩B = 2,6{ }

Question 3

The Venn diagram shows thedistribution of elements in twosets A and B. Transfer the information in the Venn diagramto a two-way table and find:(i) P A ∩B( )(ii) P A ∩B'( )

(i) n A ∩B( ) = 2

∴ P A ∩B( ) = 217

(i) n A ∩B'( ) = 3

∴ P A ∩B'( ) = 317

A A’

B 2 8 10 B’ 3 4 7

5 12 17

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SET 1 SOLUTIONS The Complement, Union and Intersection Question 4

From a group of 40 adults, 24 enjoy reading crime fiction (C), 14 enjoyreading historical fiction (H) and 6 enjoy reading crime fiction andhistorical fiction.(a) Illustrate the information on a Venn diagram.(b) Find the probability that a person chosen at random from the group will enjoy reading: (i) crime fiction (ii) only crime fiction

(i) n(C) = 24

∴ P(C)=2440

= 35

(ii) n(C only) =18

∴ P(C only) = 1840

= 920

Question 5

In a group of 26 students, 18 play tennis (T) and 12 play hockey (H). Assuming that each of the 26 students plays at least one of these sports, find the probability that a student chosen randomly from this group plays both tennis and hockey.

Since each of the 26 students plays at least one of these sports, then P(T ∪Η) =1From the Addition Theorem: P(T ∪Η) = P(T)+ P(H)− P(T ∩Η)

∴1 = 1826

+ 1226

− P(T ∩Η)

P(T ∩Η) = 1826

+ 1226

−1

= 213

∴ The probability that a student plays tennis and hockey is 2

13.

Question 6

Use the Venn diagram to find:(a) (i) P(A) (ii) P(B) (iii) P(A ∩B) (iv) P(A ∪B)(b) Hence show that: P(A ∪B) = P(A)+ P(B)− P(A ∩B)

(a) Let D = a +b + c +d

(i) P(A) = a +bD

(ii) P(B) = b + cD

(iii) P(A ∩B) = bD

(iii) P(A ∪B) = a +b + cD

(b) L.H.S. = a +b + cD

R.H.S. = a +bD

+ b + cD

− bD

= a +b + cD

= L.H.S.

∴ P(A ∪B) = P(A)+ P(B)− P(A ∩B)

21

SET 1 SOLUTIONS Conditional Probability Question 1 Find P(A/B) if P(A ∩B) = 0.1 and P(B) = 0.5.

P(A/B) = P(A ∩B)

P(B)= 0.1

0.5= 0.2

Question 2

A pair of dice are rolled. Given that the sum of the rolls is 6, find theprobability that at least one of dice rolls a 2.

Let A be the event "the sum of the rolls is 6"Let B be the event "at least one of the dice shows a 2"A = {(1,5),(5,1),(2,4),(4,2),(3,3)}B = {(1,2),(2,1),(2,2),(3,2),(2,3),(4,2),(2,4),(5,2),(2,5),(6,2),(2,6)}P(A ∩B) = {(2,4),(4,2)}

∴ P(B/A) = P(A ∩B)P(A)

= 2 365 36

= 25

Question 3

Use the Venn diagram to find:(i) n(A ∩B)(ii) n(B)

(iii) Using P(A/B) = n(A ∩B)n(B)

,find P(A/B)

(i) n(A ∩B) = 2(ii) n(B) = 2 + 6 = 8

(iii) P(A/B) = n(A ∩B)n(B)

= 28

= 0.25

22

SET 1 SOLUTIONS Conditional Probability Question 4

Use the two way table to find:(i) n(A ∩B)(ii) n(A)(iii) P(B/A)

A A’ B 2 6 8 B’ 3 5 8

5 11 16

(i) n(A ∩B) = 2(ii) n(A) = 5

(iii) P(B/A) = n(A ∩B)n(A)

= 25

Question 5

Two events, A and B are mutually exclusive. What is the value of:(i) P(A/B)(ii) P(B/A)

(i) If A and B are mutually exclusive then n(A ∩B) = 0.

P(A/B)=n(A ∩B)

n(B)= 0

n(B)= 0

(ii) Similarly P(B/A)=n(A ∩B)

n(A)= 0

n(A)= 0

Question 6

Two events A and B, are such that B is a subset of A, as shown in theadjacent diagram. Find:(i) P(A/B) (ii) P(B/A)

(i) n(B) = 7 and n(A ∩B) = 7

∴ P(A/B) = n(A ∩B)n(B)

= 77=1

(ii) n(A) = 20 and n(A ∩B) = 7

∴ P(B/A) = n(A ∩B)n(A)

= 720

= 0.35

23

SET 1 SOLUTIONS Independent Events Question 1

The probability that a person shops at Molesworth (M) is 34

and the

probability that the person is female (F) is 12

. If these events are

independent, find the probability that the customer of Molesworth is female.

P(M) = 34

and P(F) = 12

If the events are independent then P(M∩ F) = P(M)× P(F)

∴P(M∩ F) = 34× 1

2= 3

8

∴The probability that the customer of Molesworth is female is 38

.

Question 2

Given that P(A) = 14

and P(B) = 13

and P(A ∪B) = x.

Find x if :(i) A and B are mutually exclusive. (ii) A and B are independent.

Question 3

Consider the events A and B, with the number of elements contained in each event given in the adjacent Venn diagram.Given that the events A and B areindependent, find the value of x.

P(A ∩B) = P(A)× P(B)

P(A ∩B) = 2(10 + 2 + 4)+ x

= 216 + x

P(A) = (10 + 2)16 + x

= 1216 + x

P(B) = (2 + 4)16 + x

= 616 + x

∴ 216 + x

= 1216 + x

× 616 + x

2(16 + x )2

16 + x= 72

16 + x = 36∴ x = 20

(i) P(A ∪B) = P(A)+ P(B)− P(A ∩B)If A and B are mutually exclusive events then P(A ∩B) = 0∴x = P(A)+ P(B)

= 14+ 1

3

∴ x = 712

(ii) If A and B are independent events then P(A ∩B) = P(A)× P(B)

∴ P(A ∩B) = 14× 1

3

= 112

x = P(A)+ P(B)− P(A ∩B)

=14+ 1

3− 1

12

∴ x = 12

24

SET 1 SOLUTIONS Independent Events Question 4 For two independent events A and B, P(A ∪B) = 0.8 and P(B) = 0.4. Find P(A).

Let P(A ∩B) = y and P(A) = xP(A ∪B) = P(A)+ P(B)− P(A ∩B)

0.8 = x + 0.4 − yx − y = 0.4 ...(1)

P(A ∩B) = P(A)× P(B)y = x × 0.4 ....(2)

Substituting (2) into (1)x − 0.4x = 0.4

0.6x = 0.4

x = 23

∴ P(A) = 23

Question 5

For the events A and B, with details provided in the two-way table,complete the table and decide if the events A and B are independent.

A A’ B 1 2 3 B’ 3 6 9 4 8 12

Question 6

For the independent events A and B and given the variables provided in the Venn diagram, show that ac = bd.

If the events A and B are independent then:P(A ∩B) = P(A)× P(B)Let D = a +b + c +d

P(A ∩B) = bD

,P(A) = (a +b)D

, P(B) = (b + c )D

∴bD

= (a +b)D

× (b + c )D

bD = (a +b)(b + c )b(a +b + c +d) = ab +ac +b2 +bcab +b2 +bc +bd = ab +ac +b2 +bc∴ bd = ac

Using Set notation:n A ∩B( ) ×n A'∩B'( ) = n A ∩B'( ) ×n A'∩B( )

P(A) = 412

= 13

,P(B) = 312

= 14

P(A)× P(B) = 13× 1

4= 1

12

P(A ∩B) = 112

∴ P(A ∩B) = P(A)× P(B)∴ A and B are independent events.

25

SET 1 SOLUTIONS Sampling With and Without Replacement Question 1

Question 2 A bag contains 6 red balls and 4 black balls. A ball is taken and its colour noted. It is not replaced. A second ball is taken and its colour noted. Find the probability of obtaining: (i) Two red balls. (ii) Two balls of a different colour.

Question 3

A factory assembles torches. Each torch has one battery (Ba) and one bulb (B

u).

It is known that 5% of all batteries and 3% of all bulbs are defective.Find the probability that, in a torch selected at random, both the battery andthe bulb are not defective.

Let the probability of: a defective battery be P(B

a),a defective bulb be P(B

u)

a non-defective battery be P(nBa),a non-defective bulb be P(nB

u)

P(Ba) = 0.05 → P(nB

a) = 0.95

P(Bu) = 0.03 → P(nB

a) = 0.97

∴ P(both not defective) = 0.95 × 0.97= 0.9215

A die is tossed three times. What is the probability of obtaining:(i) Three sixes ?(ii) Exactly two sixes ?(ii) At least one six ?

From the tree diagram:

(i) P(3 sixes) = 1216

(ii) P(2 sixes) = 3 × 5216

= 572

(iii) P(at least one 6) =1− 125216

= 91216

From the tree diagram:

(i) P(RR) = 610

× 59= 30

90= 1

3

(ii) P(RB) = 610

× 49+ 4

10× 6

9= 48

90= 8

15

26

SET 1 SOLUTIONS Sampling With and Without Replacement Question 4

A fair 4-sided die is rolled twice.(a) List the sample space in the adjacent table (b) Find the probability of:

(i) a double (ii) a sum of at least 5 (iii) a sum not equal to 3

(i) Let D = (1,1),(2,2),(3,3),(4,4){ }∴ P(D) = 4

16= 1

4(ii) Let the most 4 be M4

M4 = (1,1),(1,2),(1,3),(2,1),(2,2),(3,1){ }Let the least 5 be L5∴ P(L5) =1− P(M4)

=1− 616

= 58

(iii) Let the sum = 3 be S3 S3 = (1,2),(2,1){ }

Let the sum ≠ 3 be (SN3)∴ P(SN3) =1− P(S3)

=1− 216

= 78

1 2 3 4 1 (1,1) (1,2) (1,3) (1,4) 2 (2,1) (2,2) (2,3) (2,4) 3 (3,1) (3,2) (3,3) (3,4) 4 (4,1) (4,2) (4,3) (4,4)

Question 5

A group of 20 children met at MacDougalls, 13 had a Big Doug,15 hada thick shake and 9 had a Big Doug and a thick shake. Complete the two-way table provided and find the probability that:(i) a child only had a Big Doug.(ii) a child had a Big Doug given that they had a thick shake.

From the table:

(i) P(BD) = 420

= 15

(ii) P(BD/TS) = n(BD∩ TS)n(TS)

= 915

= 35

BD BD’

TS 9 6 15 TS’ 4 1 5

13 7 20

Question 6 Tom (T) and Jerry (J) are playing a game of Chess. They will play two games and each has an equal chance of winning the first game. If Tom wins the first game, his probability of winning the second game is increased to 0.7. If Tom loses the first game, his probability of winning is reduced to 0.4. (i) Complete the adjacent tree diagram. (ii) Find the probability that Tom wins exactly one game.

(ii) P(Tom wins) = 0.5 × 0.3 + 0.5 × 0.4 = 0.35

27

SET 1 SOLUTIONS Discrete Probability Distributions Question 1

A fair coin is thrown twice. The random variable X is the number of headsobtained. Tabulate the probability distribution of X.

Let x = HH,HT,TH,TT{ }∴ P(x = 0) = 1

4,P(x =1) = 2

4,P(x = 2) = 1

4

x 0 1 2 Total

P X = x( )

14

12

14

1

Question 2

Two 4-sided dice are thrown simultaneously. The random variable Dis the difference between the smaller and the larger score, or zero if they arethe same. Using the data from the adjacent diagram, tabulate the probability distribution of D.

1 2 3 4 1 0 1 2 3 2 1 0 1 2 3 2 1 0 1 4 3 2 1 0

d 0 1 2 3 Total

P D = d( )

14

38

14

18

1

Question 3 In the following probability distribution, c is a constant. Find the value of c.

x 0 1 2 3

P X = x( )

13

14

15

c

P X = x( )∑ =1

∴ 13+ 1

4+ 1

5+ c =1

4760

+ c =1

∴ c = 1360

28

SET 1 SOLUTIONS Discrete Probability Distributions Question 4

The probability distribution of the random variable S is given in the following table, where c is a constant. Find the value of c.

s 1 2 3 4 5

P S = s( ) c 3c

c2 c

2 95 128

c + 3c + c 2 + c 2 + 95128

=1

2c 2 + 4c − 33128

= 0

256c 2 + 512c − 33 = 0

c =−512 ± (512)2 − 4 × 256 × −33

2 × 256

= −512 ± 295936512

c = −512 ± 544512

= −21

16 or

116

∴ c = 116

is the only valid solution. (0 ≤ c ≤1)

Question 5

Find the expected value of the variable X, which has the probability distribution given below.

x 1 2 3 4 5 6 Total

P X = x( ) 1/6 1/6 1/6 1/6 1/6 1/6 1

E(X)= xi∑ p

i= 1× 1

6

⎛⎝⎜

⎞⎠⎟+ 2 × 1

6

⎛⎝⎜

⎞⎠⎟+ 3 × 1

6

⎛⎝⎜

⎞⎠⎟+ 4 × 1

6

⎛⎝⎜

⎞⎠⎟+ 5 × 1

6

⎛⎝⎜

⎞⎠⎟+ 6 × 1

6

⎛⎝⎜

⎞⎠⎟

= 312

Question 6

Find the expected value and variance of the variable X, which has the probability distribution given below.

x 1 2 3 4 5

P X = x( ) 0.2 0.3 0.2 0.1 0.2

E(X) = xi∑ p

i= 1× 0.2( ) + 2 × 0.3( ) + 3 × 0.2( ) + 4 × 0.1( ) + 5 × 0.2( ) = 2.8

Var(X) = xi2p

i∑ − µ2 µ = E(X)( )∴ Var(X) = 12 × 0.2( ) + 22 × 0.3( ) + 32 × 0.2( ) + 42 × 0.1( ) + 52 × 0.2( ) − 2.82

=1.96