Calculate Probability Candidate Travelled by Bus Given Late Arrival

Let $B$, $S$, and $C$ denote the events that the candidate travels by bus, scooter, and car, respectively, and let $L$ denote the event that the candidate arrives late.

We are given the prior probabilities: $P(B) = \frac{2}{5}, \quad P(S) = \frac{1}{5}, \quad P(C) = \frac{2}{5}$

The conditional probabilities of being late are: $P(L|B) = \frac{1}{5}, \quad P(L|S) = \frac{1}{3}, \quad P(L|C) = \frac{1}{4}$

Using Bayes' Theorem, the probability that the candidate travelled by bus given that they arrived late is: $P(B|L) = \frac{P(B) P(L|B)}{P(B) P(L|B) + P(S) P(L|S) + P(C) P(L|C)}$

Substituting the given values into the formula: $P(B|L) = \frac{\frac{2}{5} \cdot \frac{1}{5}}{\left(\frac{2}{5} \cdot \frac{1}{5}\right) + \left(\frac{1}{5} \cdot \frac{1}{3}\right) + \left(\frac{2}{5} \cdot \frac{1}{4}\right)} = \frac{\frac{2}{25}}{\frac{2}{25} + \frac{1}{15} + \frac{1}{10}} = \frac{\frac{2}{25}}{\frac{37}{150}} = \frac{12}{37}$

Thus, the correct option is B.