A doctor is to visit a patient. It is known that the probabilities that he will come by train, bus, scooter or by other means of transport are respectively $\frac{3}{10}, \frac{1}{5}, \frac{1}{10}$ and $\frac{2}{5}$. The probabilities that he will be late are $\frac{1}{4}, \frac{1}{3}$ and $\frac{1}{12}$, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late. When he arrives, he arrives late.

The probability that he comes by bus is:

$\frac{4}{9}$

T → doctor comes by train

B → doctor comes by bus

S → doctor comes by scooter

O → doctor comes by other train

L → doctor comes late

P(T) = $\frac{3}{10}$    P(B) = $\frac{1}{5}$    P(S) = $\frac{1}{10}$    P(O) = $\frac{2}{5}$

$P(\frac{L}{T}) = \frac{1}{4}    P(\frac{L}{B}) = \frac{1}{3}    P(\frac{L}{S}) = \frac{1}{12}    P(\frac{L}{O}) = 0$

to find the probability that he comes by bus given that he is late

(Using bayes thorem)

P (B/L) = $\frac{P(B) P(L/B)}{P(T) P(L / T)+P(B) P(L / B)+P(S) P(L / S)+P(0) P(L / O)}$

$=\frac{\frac{1}{5} \times \frac{1}{3}}{\frac{3}{10} \times \frac{1}{4}+\frac{1}{5} \times \frac{1}{3}+\frac{1}{10} \times \frac{1}{12}+\frac{2}{5} \times 0}$

$=\frac{4}{9}$