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Probability I

SPM Practice (Short Questions)

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Question 6:
A box contains blue marbles and green marbles. A marble is picked at random from the box.
The probability of picking a blue marble is 6 11 . If there are 30 green marbles in the box, how many blue marbles are there in the box?

Solution:
Probability of picking a green marble = 1 – 6 11 = 5 11

Total number of marbles in the box = 30 × 11 5 = 66

Hence, number of blue marbles in the box = 66 – 30 = 36 



Question 7:
Table below shows the distribution of a group of 90 pupils attending a penalty kick training programme.

Form Four
Form Five
Girls
33
15
Boys
18
24
A pupil is chosen at random from the group to start the penalty kick training.
What is the probability that a boy from Form Five will be chosen?

Solution:
P (a boy from Form Five)
= 24 90 = 4 15


Question 8:
Diagram below shows some number cards.
 
A card is picked at random. State the probability that a prime number is picked.

Solution:
prime number = {11, 19, 37}
P (a prime number)
= 3 6 = 1 2

SPM Practice (Short Questions)

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Question 4:
John has a collection of stamps from Thailand, Indonesia and Singapore. He picks one stamp at random. The probability of picking a Thailand stamp is 1 5  and the probability of picking an Indonesia stamp is 8 15 .  John has 20 Singapore stamps. Calculate the total number of stamps in his collection. 

Solution:
Probability of picking a Singapore stamp = 1 − 1 5 − 8 15 = 4 15 Given that John has 20 Singapore stamps, ∴ Total number of stamp in John's collection = 15 4 × 20 = 75



Question 5:
A box contains 36 green erasers and a number of red erasers.
If an eraser is picked randomly from the box, the probability of picking a red eraser is 5 8 . Find the number of red erasers.

Solution:
Probability of picking a green eraser = 1 – 5 8 = 3 8

Total number of erasers in the box = 36 × 8 3 = 96

Hence, number of red erasers in the box = 96 – 36 = 60


SPM Practice (Short Questions)

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Question 1:
A box contains 5 pink marbles and 21 yellow marbles. Sharon puts another 4 pink marbles and 1 yellow marble inside the box. A marble is chosen at random from the box.
What is the probability that a pink marble is chosen?

Solution:
Total number of pink marbles = 5 + 4 = 9
Total number of yellow marbles = 21 + 1 = 22
Total number of marbles = 9 + 22 = 31

∴ P ( pink marble ) = 9 31


Question 2:
In a group of 80 prefects, 25 are girls. Then 10 boys leave the group.
If a prefect is chosen at random from the group, state the probability that the prefect chosen is a boy.

Solution:
Number of boys = 80 – 25 = 55

After 10 boys left,
Number of boys = 55 – 10 = 45

Total number of prefects = 25 + 45 = 70
∴ P ( b o y ) = 45 70 = 9 14


Question 3:
60 people have taken part in a singing contest. If a person is chosen at random from all the contestants, the probability of getting a male contestant is 8 15 . If there are 12 males and 3 females who did not qualify to the second round, find the probability that a male is chosen from the second round contestants.

Solution:
Number of male contestants = 60 × 8 15 = 32

After 12 males and 3 females left,
Total number of contestants left = 60 – 15 = 45

Number of male contestant = 32 – 12 = 20
∴ P ( male contestant ) = 20 45 = 4 9

7.3 Probability of an Event

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(A) Probability of an Event
1. The probability of an event A, P(A) is given by

2. If P(A) = 0, then the event A will certainly not occur.
3. If P(A) = 1, then the event A will certainly to occur.

Example 1:
Table below shows the distribution of a group of 80 pupils playing a game.
Form Four
Form Five
Girls
28
16
Boys
12
24
A pupil is chosen at random from the group to start the game.
What is the probability that a boy from Form Five will be chosen?

Solution:
Let
A = Event that a boy from Form Five
S = Sample space
n(S) = 28 + 12 + 16+ 24 = 80
n(A) = 24
P ( A ) = n ( A ) n ( S ) = 24 80 = 3 10



(B) Expected Number of Times an Event will Occur
If the probability of an event A and the number of trials are given, then the number of times event A occurs
= P(A) × Number of trials
 
Example 2:
In a football training session, the probability that Ahmad scores a goal in a trial is ⅝. In 40 trials are chosen randomly, how many times is Ahmad expected to score a goal?
 
Solution:
Number of times Ahmad is expected to score a goal
= ⅝ × 40
= 25  



(C) Solving Problems
Example 3:
Kelvin has 30 white, blue and red handkerchiefs. If a handkerchief is picked at random, the probability of picking a white handkerchief is 2 5 .  Calculate
(a) the number of white handkerchiefs.
(b) the probability of picking a blue handkerchief if 8 of the handkerchiefs are red in colour.
 
Solution:
Let
W = Event that a white handkerchief is picked.
B = Event that a blue handkerchief is picked.
R = Event that a red handkerchief is picked.
S = Sample space

(a)
n(S) = 30
n ( W ) = P ( W ) × n ( S ) = 2 5 × 30 = 12

(b)
Given n(R) = 8
n(B) = 30 – 12 – 8 = 10
P ( B ) = n ( B ) n ( S ) = 10 30 = 1 3

7.1 Sample Space

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7.1 Sample Space
 
(A) Experiment
1. An experiment is a process or an action in making an observation to obtain the required results.
2. The result of the experiment is called the outcome.


(B)    Listing all the possible outcomes of an experiment
We can list all the possible outcomes of an experiment by carrying out the experiment or by reasoning.

Example 1:
Six cards as shown in the diagram above are placed in a box. A card is drawn at random from the box. List all the possible outcomes.

Solution:
All the possible outcomes are O, R, A, N, G, E.


(C)   Sample space of an experiment
1. A sample space is the set of all the possible outcomes of an experiment.
2. The letter S is used to represent the sample space and all the possible outcomes are written in brackets, { }.
 
Example 2:
A letter is chosen from the word ‘GARDEN’.
(a) List all the possible outcomes.
(b) Write the sample space, S, using set notation.
 
Solution:
(a) The possible outcomes are G, A, R, D, E and N
(b) Sample space, S = {G, A, R, D, E, N }

7.2 Events

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7.2 Events
(A) Elements of a Sample Space which Satisfy Given Condition
When a specific condition is given, we can list the elements of a sample space which satisfy the given condition.
 
Example 1:
A two-digit number which is not more than 25 is chosen at random. List the elements of the sample space which satisfy each of the following conditions.
(a) A perfect square is chosen.
(b) A prime number is chosen.
 
Solution:
Sample space
= {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}

(a) 
A perfect square is chosen
= {16, 25}
 
(b)
A prime number is chosen
= {11, 13, 17, 19, 23}



(B) Events for a Sample Space
An event is a set of outcomes which satisfy a specific condition.
Event is a subset is a sample space.


Example 2:
A coin and a die are thrown simultaneously. The events P and Q are defined as follows.
P = Event of obtaining a ‘heads’ from the coin and an odd number from the die.
Q = Event of obtaining a ‘tails’ from the coin and a number more than 2 from the die.

(a) List the sample space, S.
(b) List the elements of
(i) The event P,
(ii) The event Q.
 
Solution:
(a)
Sample space, S
= { (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6) }
 
(b)(i)
P = { (H, 1), (H, 3), (H, 5) }
← (Set of outcomes of Event P: obtaining a ‘heads’ from the coin and an odd number from the die.)
 
(b)(ii)
Q = { (T, 3), (T, 4), (T, 5), (T, 6) }
← (Set of outcomes of Event Q: obtaining a ‘tails’ from the coin and a number more than 2 from the die.)