A is known to speak truth 3 times out of 5 times. He throws a die and reports that it is one. Find the probability that it is actually one.
Let us assume U1, U2 and A be the events as follows:
U1 = Getting 1 on throwing a die
U2 = Getting other than 1 on throwing a die
A = Reporting 1 after throwing the die
From the problem,
⇒
⇒
⇒ P(A|U1) = P(Reporting 1 on actually getting 1 on throwing a die)
⇒ P(A|U1) = P(Telling the truth)
⇒
⇒ P(A|U2) = P(Reporting 1 but not getting 1 on throwing a die)
⇒ P(A|U2) = P(Not telling the truth)
Now we find
P(U1|A) = P(the die actually shows 1 given that man reports 1)
Using Baye’s theorem:
⇒
⇒
⇒
⇒
∴ The required probability is .