Two people agree to meet at a coffee shop. They each independently pick a random moment in time between 8 a.m. and 9 a.m. and show up exactly at their selected time. But they are very impatient, and only stay for 10 minutes after when they arrive. What is the probability that they meet? Express your answer as a common fraction.

If person A shows up between 8:10 and 8:50, the probability of their meeting is 1/3, because person B can show up as much as ten minutes earlier or later. The probability of meeting decreases to 1/6 if person A shows up at exactly 8 or 9 AM. The situation between 8:00 and 8:10 and between 8:50 and 9:00 is a bit more complex. I suggest. you work it out.

11/36

11/36

To find the probability that the two people meet, we need to consider the time ranges within which they arrive.

Let's denote the arrival time of the first person as A and the arrival time of the second person as B.

The first person can arrive at any time between 8 a.m. and 9 a.m., which gives a total of 60 minutes for the range of A.

However, since the first person only stays for 10 minutes after arrival, the possible arrival times for A are restricted to a range of 50 minutes.

Similarly, the second person can also arrive at any time between 8 a.m. and 9 a.m., resulting in a range of B of 60 minutes.

But considering that the second person also only stays for 10 minutes after arrival, the possible arrival times for B are restricted to a range of 50 minutes.

Now, in order for the two people to meet, their arrival times need to overlap. The total probability of them meeting is equal to the ratio of the overlapping time range to the total possible time range.

To calculate the overlapping time range, we need to consider three cases:
1. A arrives before B: In this case, the range of overlapping time is 50 minutes, as B can arrive anywhere within the 50-minute range after A's arrival time.
2. A arrives after B: Similar to the first case, the range of overlapping time is 50 minutes.
3. A and B arrive at the same time: In this case, the overlapping time is simply 10 minutes, as they stay for the same duration after arrival.

The total overlapping time range is the sum of these three cases, which is (50 + 50 + 10) = 110 minutes.

The total possible time range is the product of the individual time ranges for A and B, which is (50 × 50) = 2500 minutes.

Finally, we can calculate the probability by dividing the overlapping time range by the total possible time range:

Probability of meeting = overlapping time range / total possible time range
= 110 / 2500
= 11 / 250

Therefore, the probability that the two people will meet is 11/250.