7.2 Probability of the Combination of Two Events


7.2 Probability of the Combination of Two Events
1. For two events, A and B, in a sample space S, the events AB (A and B) and A υ B (A or B) are known as combined events.

2.
The probability of the union of sets A and B is given by:

P ( A B ) = P ( A ) + P ( B ) P ( A B )

3.
The probability of the union of sets A and B can also be calculated using an alternative method, i.e.

P ( A B ) = n ( A B ) n ( S )

4.
The probability of event A and event B occurring, P(AB) can be determined by the following formula.

P ( A B ) = n ( A B ) n ( S )



Example:
Given a universal set ξ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. A number is chosen at random from the set ξ . Find the probability that
(a) an even number is chosen.
(b) an odd number or a prime number is chosen.

Solution:
The sample space, S = ξ
n(S) = 14

(a)
Let A = Event of an even number is chosen
A = {2, 4, 6, 8, 10, 12, 14}
(A) = 7
P ( A ) = n ( A ) n ( S ) = 7 14 = 1 2

(b)
Let,
B = Event of an odd number is chosen
C = Event of a prime number is chosen
B = {3, 5, 7, 9, 11, 13, 15} and (B) = 7
C = {2, 3, 5, 7, 11, 13} and (C) = 6

The event when an odd number or a prime number is chosen is B 
υ C.
(B υ C) = (B) + (C) – (B C) 
B C = {3, 5, 7, 11, 13}, ( C) = 5

P ( B C ) = P ( B ) + P ( C ) P ( B C ) = n ( B ) n ( S ) + n ( C ) n ( S ) n ( B C ) n ( S ) = 7 14 + 6 14 5 14 = 8 14 = 4 7 The probability of choosing an odd number or a prime number = 4 7 .