Q. 100

# State True or False for the statements in the Exercise.

If A and B are independent events, then P(A′ ∪ B) = 1 – P (A) P(B′)

Answer :

**TRUE**

If A and B are independent events. It implies-

P(A ∩ B) = P(A)P(B)

**∵** P(A′ ∪ B) = P(A’) + P(B) – P(A’ ∩ B)

and P(A′ ∪ B) represents the probability of event ‘only B’ excluding common points.

From Venn diagram we can see:

∴ P (A′ ∩ B) = P(B) – P (A ∩ B)

⇒ P (A′ ∪ B) = P(A’) + P(B) – P(B) + P (A ∩ B)

⇒ P (A′ ∪ B) = 1 – P(A) + P(A)P(B) {independent events}

⇒ P(A′ ∪ B) = 1 – P(A){1 – P(B)}

⇒ **P(A′** **∪** **B) = 1 – P(A)P(B’) …proved**

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