# 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 .

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