A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by train, bus , scooter or by other means of transportation are respectiely `3/10,1/5,1/10` or `2/5`. The probabilities that he will be late are `1/4,1/3` and `1/12` if he comes by train, bus and scooter respectively but if he comes by other means of transport, then he will not be late. If the arrives late, find the probability that he comes by train.

Explanation

`E` = doctor visits the patient late and `T_{1}, T_{2}, T_{3}, T_{4}` doctor comes by train, bus, scooter or taxi.


so`{P}({T}_{1})=frac{1}{10}, {P}({T}_{2})=frac{1}{5}, {P}({T}_{3})=frac{3}{10}, {P}({T}_{4})=frac{2}{5}`


`P(E | T_{1})=` Probability the doctor late comes by train `=frac{1}{4}`


`P(E | T_{2})=frac{1}{3}, {P}(E | T_{3})=frac{1}{12}, {P}({E} | T_{4})=0`


Using Baye's Theorem,
`P(T_{2} | E)=` Probability the doctor arriving late comes by bus



`=frac{P(T_{2}) P(E | T_{2})}{P(T_{1}) P(E | T_{1})+P(T_{2}) P(E | T_{2})+P(T_{3}) P(E | T_{3})+P(T_{4}) P(E | T_{4})} `<`br>

...