Page 19 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 2 Probability
Exhaustive events

Two events are said to be exhaustive if it is certain that at least one of them occurs. If the events A and B
are exhaustive, they together form the whole sample space. In set language, A  B = S.

For example, when tossing a die the events “getting an even number” and “getting an odd number” are
exhaustive, because they include all possible outcomes.

Example 21

In a group of 10 students, 5 are form-one boys, 3 are form-one girls and the remaining 2 are form-two
girls. A student is randomly chosen from the group. The events A, B, C and D are defined as follows. 2
A : The selected student is a form-one student,
B : The selected student is a girl,
C : The selected student is a boy,
D : The selected student is a form-two girl,
Identify which pairs of the events are exhaustive.
Solution: The following pairs of events are exhaustive.
• A and B, because it is certain that at least one of both events occur.
• A and D, because it is certain that at least one of both events occur.
• B and C, because it is certain that at least one of both events occur.
If the events A and B are exhaustive, then A  B = S.
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So, P(A  B) = 1.

Demonstrating
Mutually exclusive events Exclusive Events
VIDEO
We are often interested in finding the probability of events whose outcomes are described by two or more
other events. For example, a secondary school data shows that 28% of students age 13 years and 16%
age 17 years. If a student from the school is selected at random, what is the probability that the student
ages 13 years or 17 years? In the following section we expand the probability calculation to include two or
more events.

Two or more events are mutually exclusive or disjoint if the events cannot occur at the same time when the
experiment is performed. Mutually exclusive events can be shown by using a Venn diagram as follows:

S
A B

If A and B are two mutually exclusive events, then
(a) they do not have any outcomes in common or cannot both occur at the same time,
i.e. A  B = φ, the intersection of A and B is the empty set.
(b) P(A and B) = 0, i.e. P(A  B) = 0
(c) P(A or B) = P(A) + P(B), i.e. P(A  B) = P(A) + P(B).

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02 STPM Math(T) T3.indd 87 28/10/2021 10:21 AM