A ferry's random departure minute

Question

A small ferry is equally likely to leave at any instant between 12:00 and 12:40. In plain terms, its departure time is spread evenly across a 40-minute window (a uniform distribution).

What is the probability that it departs during the last 10 minutes, between 12:30 and 12:40?

Accepted Answer

When a value is equally likely to land anywhere in an interval, probability is just a matter of measuring lengths. The chance of landing in some target stretch equals the length of that stretch divided by the length of the whole window. P(	ext{event}) = \frac{	ext{length of favorable part}}{	ext{length of whole interval}}. Here the whole window is 40 minutes long, and the favorable part (12:30 to 12:40) is 10 minutes long. So P(	ext{last 10 minutes}) = \frac{10}{40} = \frac{1}{4} = 0.25. There is nothing to integrate and no bell curve to look up. Because every minute is equally likely, "h…

A ferry's random departure minute

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