# 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.

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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?

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.

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