The probabilities that A and B speak the trut | vedclass

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(D) Let $P(A)$ be the probability that $A$ speaks the truth and $P(B)$ be the probability that $B$ speaks the truth. Given,$P(A) = \frac{4}{5}$ and $P

Vedclass · Accepted Answer

$\frac{7}{20}$

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(D) Let $P(A)$ be the probability that $A$ speaks the truth and $P(B)$ be the probability that $B$ speaks the truth. Given,$P(A) = \frac{4}{5}$ and $P(B) = \frac{3}{4}$. The probability that $A$ does not speak the truth is $P(\overline{A}) = 1 - P(A) = 1 - \frac{4}{5} = \frac{1}{5}$. The probability that $B$ does not speak the truth is $P(\overline{B}) = 1 - P(B) = 1 - \frac{3}{4} = \frac{1}{4}$. They contradict each other if one speaks the truth and the other lies. Thus,the probability of contradiction is $P(A \cap \overline{B}) + P(\overline{A} \cap B) = P(A) \cdot P(\overline{B}) + P(\overline{A}) \cdot P(B)$. Substituting the values: $(\frac{4}{5} 	imes \frac{1}{4}) + (\frac{1}{5} 	imes \frac{3}{4}) = \frac{4}{20} + \frac{3}{20} = \frac{7}{20}$.

The probabilities that $A$ and $B$ speak the truth are $\frac{4}{5}$ and $\frac{3}{4}$ respectively. The probability that they contradict each other when asked to speak on a fact is

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