P(F)
(by  (1))
= P(E) (1−P(F))
= P(E) P(F′)
Hence,  E and F′ are independent
(E
∩F )’
(E
F)
’∩
E
F
S
(E
F)
∩
(E
F )
’
’
∩
Fig 13 3
© NCERT
not to be republished
 546
MATHEMATICS
�Note In a similar manner, it can be shown that if the events E and F are
independent, then
(a)
E′ and F are independent,
(b)
E′ and F′ are independent
Example 14 If A and B are two independent events, then the probability of occurrence
of at least one of A and B is given by 1– P(A′) P(B′)
Solution We have
P(at least one of A and B) = P(A ∪ B)
= P(A) + P(B) − P(A ∩ B)
= P(A) + P(B) − P(A) P(B)
= P(A) + P(B) [1−P(A)]
= P(A)  +  P(B) P(A′)
= 1− P(A′) + P(B) P(A′)
= 1− P(A′)  [1− P(B)]
= 1− P(A′) P (B′)
EXERCISE 13