1. Subtopic : 10.1 Events and Probability
LECTURE 1 OF 6
TOPIC 10 : PROBABILITY
Subtopic : Events and Probability

2. LEARNING OUTCOMES:
At the end of the lesson, students should be able to
(a) understand the concept of random experiments, outcomes, sample space, events and random selections
(b) relate events as sets and use the basic set operations to define outcome of events

3. (c) understand exclusive and exhaustive events
(d) define probability and understand some basic laws of probability

4. Suppose that there are a blue pen, a black pen and a grey pen.
What’s the chance of choosing a black pen?
Ans :
Thus, the probability of choosing a black pen is

5. The Definition of Basic Terms in Probability
(i) A random experiment ~ a process leading to at least two possible outcomes with uncertainty as to which they will occur.
(ii) Basic outcomes ~ the possible outcomes of a random experiment
(iii) Sample Space ~ the set of all basic outcomes ( denoted by S )

6. The result will be either a “ head ” or a “ tail ”
Random Experiment
Example 1
Tossing a fair coin
The result will be either a “ head ” or a “ tail ”
Basic Outcomes
Sample Space S = { H ,T }
H ~ head , T ~ tail

7. (iv) Event ~ is a subset of basic outcomes from the sample space.
Example 2
Rolling a die
the basic outcomes are the numbers 1, 2, 3, 4, 5 and 6. 2 4 6
Thus S = { 1, 2, 3, 4, 5 ,6 }
1 3 5
If A = { 1, 3, 5 }
Then A is an event of getting odd numbers

8. Example 3
List a sample space when two dice are tossed or a die is tossed twice .
S = { (1,1) , (1,2) , (1,3) , (1,4) , (1,5) , (1,6)
(2,1) , (2,2) , (2,3) , (2,4) , (2,5) , (2,6)
(3,1) , (3,2) , (3,3) , (3,4) , (3,5) , (3,6)
(4,1) , (4,2) , (4,3) , (4,4) , (4,5) , (4,6)
(5,1) , (5,2) , (5,3) , (5,4) , (5,5) , (5,6)
(6,1) , (6,2) , (6,3) , (6,4) , (6,5) , (6,6) }
Thus n(S) = 36

9. Let A be an event of getting the sum of two numbers is 6
Thus A = { (1,5) , ( 2,4) , (3,3) , (4,2) , (5,1) }
n(A) = 5
Let B be an event of getting the sum of two numbers is a multiple of 5
Thus B = { (1,4) , ( 4,1) , (2,3) , (3,2) , (5,5) , (4,6) , (6,4) }
n(B) = 7

10. Mutually Exclusive Events
Two events A and B are called mutually exclusive if there is no intersection between the two events.
A  B =
A B

11. Let A : getting even numbers B : getting odd numbers A={2,4,6}
Example 4
Rolling a die
Let A : getting even numbers
B : getting odd numbers
A={2,4,6}
B={1,3,5}
Therefore, A  B =
A B 4 6 3 5

12. Exhaustive Events Two events A and B are said to be
exhaustive events if
A A A A4 A5
A A A8

13. Example 5
(i) Let S={1,2,3,4,5,6,7,8,9,10}.
If A={1,2,3,4,5,6} and
B ={5,6,7,8,9,10},
then n (A  B) =n (S).
Therefore,
A & B are exhaustive events.

14. (ii) Let S={1,2,3,4,5,6}.
If A={2,4,6} and
B={1,3},
then n (A  B) ≠ n (S).
Therefore,
A and B are not exhaustive
events.

15. The Probability Of An Event
Definition
The probability of an event A occurring is denoted by P(A) ,where
P(A) = =

16. P(A) satisfies the following conditions
If P(A) = 1 , => the event A is a sure event
If P(A) = 0 , => the event A is an impossible event

17. The complement of an event A is denoted
as with
where P( ) or P(A’) means the probability that A doesn’t happen

18. Example 6
Two dice are tossed, find the probability
the sum of the two numbers is 8
the sum of the two numbers is a prime number
Solution :
(a) Let A be the event the sum of the two numbers is 8
A = { (2,6) , (6,2) , (3,5), (5,3) , (4,4) }
n(A) = 5 , n(S) = 36
Thus P(A) =

19. (b). Let B be the event of getting the
(b) Let B be the event of getting the sum of the two numbers is a prime number.
B = { (1,1) , (1,2) , (1,4) , (1,6) , (2,1), (2,3) , (2,5) , (3,2) , (3,4) , (4,1) ,(4,3) , (5,2) , (5,6), (6,1) , (6,5) }
n(B) = 15 , n(S) = 36
Thus P(B) =

20. Example 7
Two fair coins are tossed simultaneously.
Find the probability of getting
(a) exactly two heads (b) at least one head

21. Solution :
Die 2
Outcomes:
Die 1 H
H , H H T
H , T H
T , H T T
T , T

22. (a) S={HH,HT,TH,TT}
Let A be the event of getting 2 heads
n(S) = 4
A = {HH}
Thus P(A) =
(b) Let B be the event of getting at least one head
B = { HH,HT,TH}
Thus P(B) =

23. Venn Diagram
A Venn Diagram can also be used to solve probability problems
Example 8
There are 100 form six students , of whom 20 are studying biology, 15 are studying chemistry and 8 are studying both Biology and Chemistry. We can illustrate this in a Venn diagram.

24. P( studying both Biology and Chemistry )
Venn diagram
B ~ Biology C B
C ~ Chemistry 12 8 7 73
P( studying both Biology and Chemistry )
P( B ∩ C ) = =

25. P(study either Biology or Chemistry or both )=
= 0.27
P(B U C)

26. Exercises
1. On a single toss of one die, find the probability of obtaining
(a) a 4
(b) an odd number
(c) an even number
(d) a number less than 4
(e) a number greater than 4
(f) an odd or an even number

27. 2. In a junior school class of 28 pupils,
2. In a junior school class of 28 pupils, 7 are in both a sports team and the school band. There are 16 pupils involved in sports teams and 10 in the school band. Find the probability that a pupil chosen at random.
(a) is only in the school band
(b) is in either a sports team or the school band
(c) is in neither a sports team nor the school band

28. Conclusion 1. Mutually Exclusive Events A  B = Ø 2. Exhaustive Events
n(A  B) = n(S)

29. 3. Probability of Event A
P(A) =
0 ≤ P(A) ≤ 1
4. Complement of Event A