Samir and Naomi both arrive at a cafe at a uniformly chosen random time between 9am and 10am. What is the probability that they arrive within ten minutes of each other?
So the way that was proposed to solve this problem was to use a square and shade in the region that corresponds to the appropriate event. This makes sense but I am trying to set it up as a double integral and I'm having problems. I denote $S$ to be samir's arrival time and $N$ to be Naomi's arrival time. I make the interval $[0,60]$. So we are looking for the region where $|S-N| <= 10$. How do I setup the integral from here?
Let the arrival time of Naomi depend on the arrival time of Samir. Since they arrive within $10$ minutes of each other, Naomi has to arrive between $10$ minutes earlier and $10$ minutes later than Samir.
This corresponds to the inequality $n-10 ≤ s ≤ n+10$. When we graph it, we find that the area equals $60 \times 60 - 2 * (\frac{50 * 50}{2})$, which equals $1100$. Can you find the probability from here?