Question

It is known that a bus will arrive at random at a certain location sometime between 3:00 P.M. and 3:30 P.M. A man decides that he will go at random to this location between these two times and will wait at most 5 minutes for the bus. If he misses it, he will take the subway. What is the probability that he will take the subway?

Answer 1

The bus and man can arrive in any time between 3:00 and 3:30.

Probability of man arriving before 3:25 = 5/6 as each 5 minute interval is equally possible for his arrival.

Probability of man arriving before 3:25 and catching the bus = 5/6 * 1/6 = 5/36 (as the probability for bus to come in any 5 minute interval is 1/6).

Probability of man to come after 3:25 = 1/6

Probability of man coming after 3:25 and catching the bus = 1/6 * 1/6 *x = x/36, where x is the probability that man comes first. Since both man and bus are equally likely to come first, x = 1/2, giving required probability = 1/72.

So, Probability that man catches the bus = 5/36 + 1/72 = 11/72.

Probability that he takes subway = 1 - 11/72 = 61/72 = 0.8472.