Probability that A speaks truth is `4/5` . A coin is tossed. A reports that a head appears. The probability that actually there was head is (A) `4/5` (B)`1/2` (C) `1/5` (D) `2/5`

Explanation

Let `E_(1) and E_(2)` be the events such that
`E_(1): ` A speaks truth
`E_(2):` A speaks false
m Let X be the event that a head appears.
`P(E_(1))=4/5`
`thereforeP(E_(2))=1-P(E_(1))=1-4/5=1/5`
If a coin is tossed, then it may result in either head (H) or tail (T). The probability of getting a head is `1/2` whether A speaks truth or not. Therefore,
`P(X//E_(1))=P(X//E_(2))=1/2`
The probability that there is actually a head is given by `P(E_(1)//X).`
`P(E_(1)//X)=(P(E_(1)).P(X\\E_(1)))/(P(E_(1)).P(X\\E_(1))+P(E_(2)).P(X//E_(2)))`
`=(4/5xx1/2)/(4/5xx1/2+1/5xx1/2)=4/5`