Allen and Bethany each arrive at a party at a random time between 1:00 and 2:00. Each stays for 15 minutes, then leaves. What is the probability that Allen and Bethany see each other at the party?
This is an AoPS problem, right?
You would draw a graph and then find the area of the shaded region, which comes out to be \(\frac{7}{16}\)
+118608 I have also used a probability contour map.
The first person can arrive any time between 0 and 1 hour (after1:00)
If the second person can also arrive in this time. That is the whole shaded triangle. So that covers all the possible eventualities. Total area of the region is 0.5* 1*1 = 0.5
They wil only see each other is the later one arrives withing 15 minutes of the first one.
That is the purple region.
The orange area, were they do not see each other is easier to fine. 0.5*0.75*0.75 = 9/32
so the probability that they DO NOT see each other is \(\frac{9}{32}\div \frac{1}{2}=\frac{9}{16}\)
The probability that they DO see each other is \(1-\frac{9}{16}=\frac{7}{16}\)
