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CHAPTER 15 PROBABILITY II 15.1 Probability of an Event Important concept: , where 1. Determine the sample space for each of the following experiments. Example : a) A fair coin is tossed. S = { heads, tails} b) A pen is taken out at random from a bag containing a blue pen and a black pen. S = c) A ball is picked at random from a bag containing a blue ball, a green ball and a red ball. S = d)A card is picked from the following cards. S = e) The result of true – false question. S = f) A number is select at random from the set S = 2. Determine the probability of an event. a) A bag contains 7 red balls, 5 green balls and 6 pink balls. A ball is picked at random. Find the probability for each of the following events. i)A = a red ball is picked n(A) = 7, n(S) = 18 ii) B = a green ball is picked n(B) = n(S) = P(B) = iii) C = a pink ball is picked n (C) = n(S) = P(C) = b) A bag contains 6 i)Red ii) white iii) green Probability II 1 H E B A T

Chapter 15 II Probability II ENHANCE

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Page 1: Chapter 15 II Probability II ENHANCE

CHAPTER 15 PROBABILITY II

15.1 Probability of an EventImportant concept:

, where

1. Determine the sample space for each of the following experiments.

Example :a) A fair coin is tossed. S = { heads, tails}

b) A pen is taken out at random from a bag containing a blue pen and a black pen.

S =

c) A ball is picked at random from a bag containing a blue ball, a green ball and a red ball.

S =

d) A card is picked from the following cards.

S =

e) The result of true – false question.

S =

f) A number is select at random from the set

S =

2. Determine the probability of an event.

a) A bag contains 7 red balls, 5 green balls and 6 pink balls. A ball is picked at random. Find the probability for each of the following events.

i) A = a red ball is picked

n(A) = 7, n(S) = 18

ii) B = a green ball is pickedn(B) =n(S) =

P(B) =

iii) C = a pink ball is pickedn (C) =n(S) =

P(C) =

b) A bag contains 6 white marbles, 8 red marbles and 12 green marbles. A marble is picked at random. Find the probability that the marble is

i) Red

n ( R) =

n(S )=

P® =

ii) whiten(W) =n(S) =

P(W)=

iii)green

c) The table shows a set of numbers. A number is chosen at random from the set.

10 11 1517 22 1916 25 18

i) Probability that a prime number is chosen=

ii) Probability that the number chosen is divisible by 2

=

iii) Probability that a multiple of 5 is chosen

=

Probability II 1

H E B A T

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d) Each vowel in the word ‘PROBABILITY’ is written on a pink card and each consonant is written on a blue card. A card is drawn at random from these cards. Find the probability that

i) a pink card is drawn ii) A vowel is drawn iii) a letter B is drawn

e)A letter is selected at random from the word ‘ASSOCIATIONS’. Find the probability of selecting

i) the letter ‘S’ ii) the letter ‘O’ The letter ‘P’

f) A factory has 24 male and 16 female workers. If a worker is selected at random, find the probability that

i) a male worker is selected

ii) a female worker is selected

iii) If 4 female workers leave the factory and, then, a worker is selected, find the probability that a female worker is selected.

3. Determine the number of objects.

Example:a) A bag contains 36

cards. The probability of getting a blue card from

the bag is . Number of

blue cards in the bag

=

b) A bag contains 48 cards. The probability of getting a green card

from the bag is .

Number of green cards in the bag=

c) A bag contains 27 cards. The probability of getting a pink card from

the bag is . Number of

pink cards in the bag=

d) A bag contains 60 cards. The probability of getting a red card

from the bag is .

Number of red cards in the bag=

e) A bag contains 27 blue pens and some black pens. The probability of picking a

blue pen is .

Total number of pens in the bag

=

f) A bag contains 20 red pens and some green pens. The probability of picking a red pen is

.

Total number of pen in the bag=

g) A bag contains 18 pink pens and some blue pens. The probability of

picking a blue pen is .

Total number of pen in the bag =

h) A bag contains 24 brown pens and some yellow pens. The probability of picking a brown

pen is . Total

number of pen in the bag =

Probability II 2

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i) A class has 12 boys and some girls. The probability of choosing a

boy is .

Total number of students in the class=

j) A factory has 21 male workers and some female workers. The probability of choosing a male

worker is .

Total number of workers in the factory =

k) There are 60 yellow files and some brown files in a drawer. The probability of choosing

a yellow file is .

Total number of files in the drawer=

l) A bag contains 25 red apples and some green apples. The probability of choosing a red

apple is .

Total number of apples in the bag=

15.2 Complement of an Event

The complement of an event A is denoted by A'. A' contains all outcomes of the sample space that are not in A.

P(A) + P(A') = 1 P(A') = 1 – P(A)

1. State the complement of each of the following events in words and set notation.Event Complement of event in

words Complement of event in set notation

a) A fair coin is tossed A = { head } A' = A head is not obtained

A' = { tail }

b) A fair dice is tossed B = An odd number is obtained

B’ =

c) A letter is picked at random from the word ‘DILIGENT’

C = The letter I is obtained

C’ =

2. Solve problems involving complement.

a) The probability that a team

loses in a match is .

What is the probability that the teams win the match?

b) The probability that Rahim is late

for school is .

What is the probability that he is on time?

c) A basket contains some red apples and some green apples. The probability of picking a

red apple is .

The probability of picking a green apple,P(Green) =

d)A tray contain some soft-boiled eggs and some hard boiled eggs. The probability of choosing a soft-

boiled egg is .

P( Hard-boiled egg) =

e) A box contains a total of f) A box contains a f) There g) A fair dice is

Probability II 3

Page 4: Chapter 15 II Probability II ENHANCE

24 red and purple grapes. The probability of choosing a

red grape is .

i) P(Purple) =

ii) Number of purple grape=

total of 40 green and purple marbles. The probability of choosing a green

marble is .

i) P(Purple) =

ii) Number of green marbles=

are a total of 64 orange and pink plates in a basin. The probability picking an

orange plate is .

i) P(pink plate) =

ii) Number of pink plate =

rolled. Find the probability thati) the number obtained is less than 5

ii) the number obtained is not less than 5

15.3 Combined Event

A combination of event A or event B is P(A or B) = P(A B)

=

A combination of an event A and event B is . P(A and B) = P(A B)

=

1.a) A fair coin and a dice are tossed. List the possible outcomes for the following event.

A = { head or number 6 is obtained }A = { (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 6) }

b) A fair coin and a dice are tossed. List the possible outcomes for the following event.

B = { tail or number 2 is obtained }B=

c) A fair coin and a dice are tossed. List the possible outcomes for the following event.

C = { number 2 or 5 is obtained }C =

d) A fair coin and a dice are tossed. List the possible outcomes for the following event.

D = { number 3 or 6 is obtained }D =

2. A fair coin is tossed twice. List the possible 3. Let set A = {P, Y, R, A, M, I, D} and set B = {3,

Probability II 4

Page 5: Chapter 15 II Probability II ENHANCE

outcomes for the following events. a) two heads are obtained b) two tails are obtained c) a head and a tail is obtained

4}. An element is chosen at random from each set. a) List the sample space, S, for the experiment. b) List the possible outcomes for the following events.

- a vowel and number 4 are chosen- a consonant and number 3 are chosen

4. A number is chosen at random from the numbers 15 to 30. List the outcomes of the combined events below. a) a multiple of 5 or a prime number b) a number that when divided by 3, has 1 or

2 as reminder   c) an even number or number 15.

5. A number is selected at random from the set List the outcomes

of the event that an even number and perfect square are selected.

Let A = an even number is selected = B = a perfect square is selected =Q = the event that an even number and perfect square are selected = =

Probability II 5

Page 6: Chapter 15 II Probability II ENHANCE

Based on the lists of possible outcomes in the sample space and the respective events, find the probability of the following combined events.

1.a) A fair coin and a dice are tossed. List the possible outcomes for the following event.

A = { head or number 6 is obtained }A = { (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 6) }

S ={(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,1), (T,1), (T,1), (T,1), (T,1)}

b) A fair coin and a dice are tossed. B = { tail or number 2 is obtained }

P(B)=

c) A fair coin and a dice are tossed. C = { number 2 or 5 is obtained }

P(C) =

d) A fair coin and a dice are tossed. D = { number 3 or 6 is obtained }

P(D) =

2. A fair coin is tossed twice. a) A = two heads are obtained b) B = two tails are obtained c) C = a head and a tail is obtained

3. Let set A = {P, Y, R, A, M, I, D} and set B = {3, 4}. An element is chosen at random from each set. a) a vowel and number 4 are chosen b) a consonant and number 3 are chosen

4. A number is chosen at random from the numbers 15 to 30. a) a multiple of 5 or a prime number b) a number that when divided by 3, has 1 or 2 as reminder   c) an even number or number 15.

5. A number is selected at random from the set

Find the probability that an even number and perfect square are selected.

Probability II 6

Page 7: Chapter 15 II Probability II ENHANCE

15.4 Questions Based on Examination Format 1.

District Residential School Non-residential schoolP 2 3Q 3 4R 5 2

The table above shows the number of residential and non-residential schools in three districts P, Q and R. The probability that Liana will be sent to district P, Q, or R is the same. Calculate the probability that Liana will be sent to a) a residential school in district Q

b) a non-residential school

2. Hassan and Lee represented their school in either a quiz or a debate. The probabilities that

Hassan and Lee represent the school in a quiz are and respectively. Calculate the

probability that a) Hassan represents the school in a quiz and Lee represents the school in the debate, b) Hassan and Lee represented the school in the same competition.

3.

Nine alphabetic cards above are placed in a box. Two cards are picked at random from the box, one after another without replacement. Calculate the probability that

a) The first card is “O” and the second card is not “O”. b) Both the cards picked have the same alphabet.

4.

The table above shows the number of black cards and white cards in box E and box F. Rose takes a card randomly from box E. If the card taken is black, the card is returned to the box. If the card is white, the card is put into box F. Then, a second card is taken from box F. Calculate the probability that Rose takes

a) Black card from box E followed by a white card from box F, b) A black card from box F.

5. In a survey carried out on a group of 18 arts enthusiasts. 10of them like music and the rest like painting. Two people are chosen at random from the group. Calculate the probability that

a) Both of them like musicb) Both of them have the same interest.

Probability II

Box E Box FBlack card 4 7White card 3 2

7

C O U R T E O U S

Page 8: Chapter 15 II Probability II ENHANCE

6. Zul and Sidek played two games. The probability that Zul wins the first game is . The

probability that Zul wins the second game is . The results of the two games cannot end in a

draw. Calculate the probability thata) Zul wins both the gamesb) Zul wins any of the games.

7. During the coming school holiday, the probabilities of Liana visiting Cameron Highlands

and Genting Highlands are and respectively. Calculate the probability that Liana

a) visits Cameron Highlands and Genting Highlandsb) visits Cameron Highlands or Genting Highlands

8. Sue and Liza choose to register in either school A or school B. The probabilities that Sue and

Liza choose school A are and respectively. Calculate the probability that

a) Sue chooses school A and Liza chooses school B.b) Sue and Liza choose the same school.

9.

The table above shows the probabilities that Zul and Hamdan will join the Science stream or the Arts stream after completing form three. Calculate the probability thata)Zul and Hamdan will join the Science streamb) one of them joins the Science stream and the other joins the Arts stream.

10.

10.

The cards above are put into a box. A card is drawn at random from the box. If the card drawn is T, the card is replaced. If the card is not T, the card is not replaced. Then, a second card is drawn at random from the box. Calculate the probability thata)both the card drawn are marked Tb) one of the card is marked T

15.5 Past Year SPM Questions

Probability II

Student The probability of joining the

Science stream Arts stream.Zul

Hamdan

8

T A A B ET T

Page 9: Chapter 15 II Probability II ENHANCE

November 2003

1. Diagram 5 shows the route of a vehicle which carries a group of volunteers. The group consists of 7 males and 5 females who are dropped off at random to sell flags at various points along the routes.

a) If two volunteers are dropped off at Taman Aman, calculate the probability that both are males.b) Two volunteers of different gender are dropped off at Taman Aman. If two other volunteers are then dropped off at Taman Sentosa, calculate the probability that at least one of them is female.

July 2004

2. Table 1 shows the number of pupils attending a motivation camp.

SchoolNumber of Pupils

Boys GirlsSchool A 8 7School B 7 3

Two pupils from the group are required to take part in a show.a) If a pupil is chosen at random from school A and another one from school B, calculate

the probability that both are girls.b) If two pupils are chosen at random from the group of boys, calculate the probability that

both are from the same school.

November 2004

3. Table 1 shows the number of coupons in two boxes, A and B. The coupons are of various values, RM1, RM2 and RM5. Students are given coupons as an incentive for selling bookmarks.

Number of CouponsBox RM1 RM2 RM5A 1 6 8B 2 5 3

Students selling more than 100 bookmarks are given the chance to draw at random a coupon from box A. Students selling less than 100 bookmarks are given the chance to draw at random a coupon from box B. Ali sells 120 bookmarks. Lim sells 52 bookmarks. Find the probability thata) Both of them draw a RM5 coupon.b) The total value of the two coupons drawn by them is less than RM4.

Probability II 9

Taman Aman Other places

Starting pointTaman Sentosa

[5 marks]

[5 marks]

[5 marks]

Page 10: Chapter 15 II Probability II ENHANCE

July 2005

4. Diagram below shows the route of a cross-country event of a school.

A group of 8 boys and 6 girls from the St. John Ambulance Brigade have been chosen for duty for the event. All of them will be taken by a van to be placed for duty at various stations. The van travels to station A, then Station B, then station C, and so on.a) Two pupils from a group are chosen at random for duty at Station A. Calculate the probability that both are girls.b) Two girls from the group are placed for duty at Station A. Two other pupils are then chosen at random for duty at Station B.

Calculate the probability that they are of different gender to each other.

November 2005

5. A group of 5 boys and 4 girls take part in a study on the type of plants found in a reserved forest area. Each day, two pupils are chosen at random to write the report.a) Calculate the probability that both pupils chosen to write the report on the first day are boys.b) Two boys do write the report on the first day. They are then exempted from writing the report on the second day. Calculate the probability that both pupils chosen to write the report on the second day are of the same gender.

July 2006

6. Table below shows the number of teachers in a two-session school.

SessionNumber of teachers

Men WomenMorning 6 10

Afternoon 4 8

Two teachers from the school are chosen at random to attend an assembly of Teacher’s Day at the state level.Calculate the probability that both teachers chosena) are men,b) are from the same session

7. November 2006

Probability II 10

Start/ End

Station A Station B

Station C

Station DStation E

[5 marks]

[5 marks]

[5 marks]

Page 11: Chapter 15 II Probability II ENHANCE

In a quiz contest, there are three categories of questions consisting of 5 questions on sports, 3 questions on entertainment and 7 questions on general knowledge.Each question is placed inside an envelope. All of the envelopes are similar and put inside a box.All the participants of the quiz contest are requested to pick at random two envelopes from the box.Find the probability that the first participant picksa) the first envelope with a sport question and the second envelope with an entertainment question, b) two envelopes with questions of the same category.

8. June 2007

Table 1 shows the number of a group of students in Form 1 Alpha and Form 1 Beta who are entitled to receive school bags.

Form

Gender1 Alpha 1 Beta

Boys 3 6Girls 5 2

Table 1 Two students from the group are chosen at random to receive a school bag each. Find the probability that both students chosen A are boys B are girls from the same class [5 marks] 9. November 2007 Diagram 4 shows ten labeled cards in two boxes.

Box P Box Q Diagram 4 A card is picked at random from each of the boxes. By listing the outcomes, find the probability that A both cards are labeled with a number B one card is labeled with a number and the other card is labeled with a letter. [5 marks]

Probability II 11

[5 marks]

A B2 C D E3 4 F G

Page 12: Chapter 15 II Probability II ENHANCE

10. June 2008 Two students are chosen at random to arrange books in the library. They are chosen from a group of 2 boys and 3 girls.By listing the sample space of the possible outcomes of the events, find the probability that

a) two girls are chosen,b) at least one boy is chosen.

11. November 2008Diagram 10 shows three numbered cards in box P and two cards labeled with letters in box Q.

P Q Diagram 10

A card is picked at random from box P and then a card is picked at random from box Q.By listing the sample of all possible outcomes of the event, find the probability that

a) a card with an even number and the card labeled Y are picked,b) a card with a number which is multiple of 3 or the card labeled R are picked. [5 marks]

Probability II 12

2 63 Y R

Page 13: Chapter 15 II Probability II ENHANCE

ANSWER

Chapter 15 Probability

15.4 Questions according to Examination Format

1

2

3

4

5

6

7

8

9

Probability II 13

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10

15.5 SPM Past Year Questions

SPM 2003

SPM 2004 J

SPM 2004

SPM 2005 J

SPM 2005

SPM 2006 J

SPM 2006

SPM 2007 J 8a) 8b)

SPM 2007 N 9a) 9b)

SPM 2008 J 10) S =

10a) 10b)

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Page 15: Chapter 15 II Probability II ENHANCE

SPM 2008 N 11) S =

11a) 11b)

Probability II 15