# In a family, the husband tells a lie in 30% cases and the wife in 35% cases. Find the probability that both contradict each other on the same fact.

Given:

Husband tells lies in 30% of cases

Wife tells lies in 35% of cases

Now,

P(NH) = P(Husband telling lies) = 0.30

P(TH) = P(Husband telling truth) = 1-0.30

P(TH) = 0.70

P(NW) = P(Wife telling lies) = 0.35

P(TW) = P(Wife telling truth) = 1-0.35

P(TW) = 0.65

We need to find the probability for case in which husband and wife contradict each other for narrating a fact.

This happens only when husband not telling truth while Wife is telling truth and vice-versa.

P(CHW) = P(Husband and Wife contradict each other)

P(CHW) = P(Husband tells truth and Wife doesn’t) + P(Wife tells truth and Husband doesn’t)

Since the speaking of husband and Wife are independent events their probabilities will multiply each other.

P(CHW ) = (P(TH)P(NW)) + (P(NH)P(TW))

P(CHW) = (0.70×0.35) + (0.30×0.65)

P(CHW) = 0.245 + 0.195

P(CHW) = 0.44

The required probability is 0.44.

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