A person goes to work by car, train or bus. The probabilities of person travelling by there modes are $3 / 10, 1 / 2$ and $1 / 5$ respectively. It is found that chance of person being late for work are $3 / 7, 1 / 7$ and $5 / 7$ by respective modes. Given that on particular day person reaches late, find the probability that on that day he had travelled by train.

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Solution

$U s i n g B a y e^{'} s R u l e P r o b a b i l i t y t h a t h e t r a v e l l e d b y t r a i n g i v e n h e i s l a t e = \frac{P r o b a b i l i t y h e i s l a t e a n d u s e d t r a i n}{P r o b a b i l i t y t h a t h e i s l a t e b y o t h e r m o d e o f t r a n s p o r t} = \frac{(1 / 2) \times (1 / 7)}{(3 / 10) \times (3 / 7) + (1 / 2) \times (1 / 7) + (1 / 5) \times (5 / 7)} = \frac{5}{24}$

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