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(AB)= 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(AB)= n(AB) n(S)   

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

  P(AB)= n(AB) 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}
n(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 n(B) = 7
C = {2, 3, 5, 7, 11, 13} and n(C) = 6

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

P(BC) = P(B)+P(C)P(B  C) = n( B ) n( S ) + n( C ) n( S ) n(BC) 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 .