A, B, and C are independent witness of an event which is known to have occurred. A speaks the truth three times out of four, B four times out of five and C five times out of six. What is the probability that the occurrence will be reported truthfully by majority of three witnesses?
Given:
A speaks truth three times out of four
B speaks truth four times out of five
C speaks truth five times out of six
⇒ P(TA) = P(A speaks truth)
⇒ 
⇒ P(TB) = P(B speaks truth)
⇒ 
⇒ P(TC) = P(C speaks truth)
⇒ 
⇒ P(NA) = P(A speaks lies)
⇒ 
⇒ 
⇒ P(NB) = P(B speaks lies)
⇒ 
⇒ 
⇒ P(NC) = P(C speaks lies)
⇒ 
⇒ 
.
It is told that the occurrence is will be reported by majority of witness. This is only possible when at least two of the witnesses speaks truth.
⇒ P(M) = P(TATBNC) + P(TANBTC) + P(NATBTc) + P(TATBTC)
Since speaking by a person is independent the probabilities will multiply each other.
⇒ P(M) = (P(TA)P(TB)P(NC)) + (P(TA)P(NB)P(TC)) + (P(NA)P(TB)P(Tc)) + (P(TA)P(TB)P(TC))
⇒ 
⇒ 
⇒ 
.
∴ The required probability is 
.