Question

Two friends arrive at a hotel independently. Each man arrives a random time between 5pm and 6pm, distributed uniformly (no moment in this time frame is any more likely for arrival than another). What do you think is the probability that the two friends arrived within 10 minutes of one another?
i) 1/36 ii) 11/36 iii) 13/36 iv) 1/3

Answer 1

1/3

Answer 2

Let x be the number of minutes after 5pm that friend 1 arrives
Let p be the probability that friend 2 arrives within 10 minutes of x
Let q be probability of friend 1 arriving on any given minute
Let P be probability that both friends arrived within 10 minutes for all possible x
\[q=\frac{1}{60}\]
\[P = \sum_{x=0}^{60}p*q = \frac{1}{60}\sum_{x=0}^{60}p = \frac{2}{60}\sum_{x=0}^{10}(\frac{1}{6}+\frac{x}{60}) + \frac{1}{60}\sum_{x=10}^{50}\frac{1}{3}\]
\[P=\frac{1}{30}\int\limits_{0}^{10}\frac{1}{6}+\frac{x}{60}dx + \frac{1}{60}\int\limits_{10}^{50}\frac{1}{3}dx\]
\[P=\frac{1}{30}(\frac{10}{6}+\frac{100}{120}) + \frac{1}{60}(\frac{40}{3})\]
\[P=\frac{1}{12} + \frac{2}{9}\]
\[P = \frac{11}{36}\]