Page 22 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 2 Probability
The events linked to an experiment may not be mutually exclusive. Consider the following case:
Suppose 10 students were asked what sports they have participated recently, their answers showed that 5
students played football and 7 students played badminton. What is the probability that one of the students
selected has played football or badminton? The probability of selecting a student who has played football is
5 or 0.5 and the probability of a student playing badminton is 7 or 0.7. If the addition law for mutually
10 10
exclusive events is used, the sum of these two probabilities is 1.2. We know that the value of probability
cannot exceed 1. So, let us check the counting. To count all the students playing either football or badminton,
we need to count all the students playing football, all the students playing badminton and subtract from this
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the number of students who were counted twice because they were playing both football and badminton.
2 To help us visualise the above reasoning, the Venn diagram is used. Assume that 3 students played both
football and badminton in the above example. The diagram indicates clearly why 3 outcomes in the overlap
area of the event A  B are being counted twice – once in A and once in B. Where event A represents “a
selected student playing football” and event B represents “a selected student playing badminton”.

A B

For events that are not mutually exclusive, the addition law is modified to take into account of double
counting. If the number of outcomes in event A is n(A) and the number of outcomes in event B is n(B),
then to find the number of outcomes in event A  B, we must count the outcomes in A  B. So, to get
the correct total without counting the outcomes in the overlap A  B twice, we must subtract the number
of outcomes in A  B. Thus
n(A  B) = n(A) + n(B) – n(A  B)

Let S be a sample space with n(S) possible outcomes. The probability of event A  B is given by

P(A  B) = n(A  B)
n(S) S
A B
= n(A) + n(B) – n(A  B)
n(S)

= n(A) + n(B) – n(A  B)
n(S) n(S) n(S)
= P(A) + P(B) – P(A  B)

This is called addition rule of probability:

P(A  B) = P(A) + P(B) – P(A  B)

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02 STPM Math(T) T3.indd 90 28/10/2021 10:21 AM