Page 32 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 2 Probability

P(R) = 0.25,
P(R) = 1 – 0.25
= 0.75
P(A | R) = 0.08, P(A | R) = 0.03.
The probability of a driver will not involve in an accident on raining tomorrow,
P(A  R) = P(R) × P(A | R)
= 0.25 × 0.92
= 0.23
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The probability of a driver will not have an accident on not-raining tomorrow,
P(A  R) = P(R) × P(A | R)
2 = 0.75 × 0.97
= 0.728

The probability that a driver will not have an accident tomorrow,
= P(A  R) + P(A  R)
= 0.23 + 0.728
= 0.958

Rule of total probability

Suppose that a sample space consists of three exhaustive and mutually S A 1 A 2 A 3
exclusive events, A , A and A . By definition, the three events do not
2
1
3
overlap and they occupy the entire sample space. The Venn diagram on
the right displays the events A , A and A and any event B.
1 2 3
B
From the diagram, the event B is composed of three mutually exclusive events A  B, A  B and A  B.
3
1
2
So, P(B) = P(A  B) + P(A  B) + P(A  B)
1 2 3
By applying the conditional probability formula to each term on the right hand side of this equation, we obtain
P(B) = P(A ) × P(B | A ) + P(A ) × P(B | A ) + P(A ) × P(B | A )
1 1 2 2 3 3
This formula is known as the rule of total probability. This rule states that the whole is the sum of its parts.

In general, for some positive integer k, let A , A , …, A be such that
1 2 k Law of

1. S = A  A  …  A k Total

1
2

2. A  A = 0 if i ≠ j INFO Probability
j
i
Then the collection of sets {A , A , …, A } is said to be a ‘partition’ of S.
1 2 k
Note: If B is any subset of S, and{A , A , …, A } is a partition of S, B can be decomposed as follows:
1 2 k

B = (A  B)  (A  B)  …  (A  B)
k
2
1
Thus, the rule of total probability states that: if {A , A , …, A } is a partition of S such that
1
2
k
P(A) . 0 for i = 1, 2, …, k, then for any event B
i
k
P(B) = ∑ P(A) P(B | A)
i = 1 i i
100
02 STPM Math(T) T3.indd 100 28/10/2021 10:21 AM