Probability that A speaks truth is \(\frac{4}{5}\). He tosses a coin and reports that a head appears. The probability that actually there was a head, is

\(\frac{4}{5}\)

Let E₁​ and E2​ be the events such that

E₁​:A speaks truth

E₂​:A speaks false

Let X be the event that a head appears.

$P(E_1​)=\frac{4}{5}$​

$∴P(E_2​)=1−P(E_1​)=1−\frac{4}{5}​=\frac{1}{5}$

If a coin is tossed, then it may result in either head (H) or tail (T).

only there will be two outcomes.

The probability of getting a head is $\frac{1}{2}$​ whether A speaks truth or not.

$∴P(\frac{X}{E_1}​)=P(\frac{X}{E_2}​​)=\frac{1}{2}$

The probability that there is actually a head is given by P(E₁/X).

$∴P(E_1/X)=\frac{P(E_1).P(X/E_1)}{P(E_1).P(X/E_1)+P(E_2).P(X/E_2)}$

$=\frac{\frac{4}{5}×\frac{1}{2}}{\frac{4}{5}×\frac{1}{2}+\frac{1}{5}×\frac{1}{2}}=\frac{\frac{1}{2}×\frac{4}{5}}{\frac{1}{2}(\frac{4}{5}+\frac{1}{5})}⇒\frac{\frac{4}{5}}{\frac{5}{5}}=\frac{4}{5}$