A friend is coming to visit Suppose the probability he comes

A friend is coming to visit. Suppose the probability he comes by tain is 1/6, by ship 1/6, by car 1/6 or by plane 1/2. If the probability that he will be late is 1/4 by train, 1/3 by ship, 1/12 by car and 0 by plane

A) Suppose we knew he was late. What is the probability that he came by train? P(Train given Late)

B) Suppose we knew he was not late. What is the probability that he came by train? P(Train given not Late)

Solution

Let

T = train
S = ship
C = car
P = plane
L = late

Thus,

a)

P(T|L) = P(T) P(L|T) / P(L)

as

P(L) = P(T) P(L|T) + P(S) P(L|S) + P(C) P(L|C) + P(P) P(L|P)

= (1/6)*(1/4) + (1/6)*(1/3) + (1/6)*(1/12) + (1/2)*0

P(L) = 1/9

thus,

P(T|L) = P(T) P(L|T) / P(L)

= (1/6)*(1/4) / (1/11)

= 11/24 [ANSWER]

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B)

P(T|L\') = P(T) P(L\'|T) / P(L\')

As

P(L\') = 1- P(L) = 1 - 1/9 = 8/9

then

P(T|L\') = (1/6)*(1-1/4)/(8/9) = 9/64 [ANSWER]