Q.13   Probability that A speaks truth is $\frac{4}{5}$ . A coin is tossed. A reports that a head appears. The probability that actually there was head is               (A)  $\frac{4}{5}$               (B)  $\frac{1}{2}$                C)  $\frac{1}{5}$               (D)  $\frac{2}{5}$

S seema garhwal

Let   A : A speaks truth

B : A  speaks false

$P(A)=\frac{4}{5}$

$P(B)=1-\frac{4}{5}=\frac{1}{5}$

X : Event that head appears.

A coin is tossed , outcomes are head or tail.

Probability of getting head whether A speaks thruth or not is $\frac{1}{2}$

$P(X|A)=P(X|B)=\frac{1}{2}$

$P(A|X)= \frac{P(A).P(X|A)}{P(B).P(X|B)+P(A).P(X|A)}$

$P(A|X)= \frac{\frac{4}{5}\times \frac{1}{2}}{\frac{4}{5}\times \frac{1}{2}+\frac{1}{5}\times \frac{1}{2}}$

$P(A|X)= \frac{\frac{4}{5}}{\frac{4}{5}+\frac{1}{5}}$

$P(A|X)= \frac{\frac{4}{5}}{\frac{1}{1}}$

$P(A|X)={\frac{4}{5}}$

The probability that actually there was head is $P(A|X)={\frac{4}{5}}$

Hence, option A is correct.

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