A person goes to his office by using different means of transport on different days. It is know that the probabilities that he will come by train, bus, scooter or by car are respectively `1/10, 3/10, 2/10 " and " 4/10`. The probabilities that he will be late are `1/4, 1/5, 1/6, " and " 1/3` if he comes by train, bus, scooter and car respectively, on one day when he reaches office, he is late. What is the probability he has come by train?

Explanation

Let `E_1, E_2, E_3 " and " E_4` be events that person come by train, bus scooter and car respectively. Then,
`P(E|E_1)` = probability that the person is late, given that he comes by train.
`1/4`
`P(E|E_2)` = probability that the person is late, given that he comes by bus
`=1/5`
`P(E|E_3)` = probability that the person is late, given that he comes by scooter
`1/6`
`P(E|E_4)`= probability that the person is late, given that he comes by car
`=1/3`
Probability that he comes by train, given that he is late
`=P(E_1|E)`
`=(P(E_1)P(E|E_1))/(P(E_1)P(E|E_1)+P(E_2)P(E|E_2)+P(E_3)P(E|E_3)+P(E_4)P(E|E_4))`
`=(1/10xx1/4)/(1/10xx1/4xx3/10xx1/5+2/10xx1/6+4/10xx1/3)`
`=15/91`