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One of the cleanest geometric probabilities: social intuition says one thing and the square of areas says another.
Two friends meet to meet between 6:00 and 7:00. Each one arrives at random at some point during that hour.
Whoever arrives first will wait exactly 15 minutes; If the other does not appear in that time, he leaves. Are they more likely to see each other or not?
What is the probability that they will be found?
Let's represent the arrival of the first friend by x and that of the second by and, both between 0 and 60 minutes. Each possible pair (x,and) corresponds to a point on a square with side 60.
They are found if and only if the difference between both arrivals is 15 minutes or less. That describes a band around the diagonal of the square.
The complementary zone is two right triangles in the corners, each with legs of 45. Between the two they occupy an area of 2025.
The total area of the square is 3600. Therefore, the probability of not being found is 2025 / 3600 = 9/16, and the probability of being found is 1 - 9/16 = 7/16.
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