My friend and I are hoping to meet for lunch. We will each arrive at our favorite restaurant at a random time between noon and 1 p.m., stay for 15 minutes, then leave. What is the probability that we will meet each other while at the restaurant? (For example, if I show up at 12:10 and my friend shows up at 12:15, then we’ll meet; on the other hand, if I show up at 12:50 and my friend shows up at 12:20, then we’ll miss each other.)
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According to what random distribution do you arrive in a given interval? Is it uniform? – user139388 May 18 '14 at 08:23
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Let $T_1$ be the time you arrive, $T_2$ the time your friend arrives, and suppose $T_1,T_2 \sim U(0,1)$ are independent. Then the probability of a meeting is \begin{align} P[T_2 \in (T_1, \min\{T_1 + 1/4,1\})] &= {\bf E}[P[T_2 \in (T_1, \min\{T_1 + 1/4,1\}) \mid T_1]] \\ &= {\bf E}\left[ \frac{1}{4} \chi\{ T_1 \leq 3/4\} + (1-T_1)\chi\{ T_1 > 3/4\} \right] \\ &= \int_0^{3/4} \frac{1}{4} du + \int_{3/4}^1 (1-u)du \\ &= \frac{3}{16} + \frac{1}{4} + \frac{1}{16} \\ &= \frac{1}{2}. \end{align}
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