Question

Probability that A speaks truth is \(\frac{4}{5}\).A coin is tossed.A report 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. (\(\frac{2}{5}\))

Answer 1

The Correct Option is A

Let A be the event that the man reports that head occurs in tossing a coin and let E1 be the event that head occurs and E2 be the event head does not occur.
P(E₁₎=\(\frac{1}{2}\),P(E₂)=\(\frac{1}{2}\)
P(A|E₁)=P(A reports that head occurs when head had actually occur red on the coin)=\(\frac{4}{5}\)
P(A|E₂)=P(A reports that leads occurs when head had not occur red on the coin)=1-\(\frac{4}{5}\)=\(\frac{1}{5}\)
By Bayes'theorem,
P(E₁|A)=\(\frac{P(E1)P(A|E_1)}{P(E_1)P(A|E_1)+P(E_2)P(A|E_2)}\)=\(\frac{\frac{1}{2}×\frac{4}{5}}{\frac{1}{2}×\frac{4}{5}+\frac{1}{2}×\frac{1}{5}}\)=4/4+1=4/5
Hence,option (A) is correct.