You and a friend have arranged to meet at a popular downtown mall between 3 p.m. and 4 p.m. one afternoon. However, you neglected to specify a time within that one-hour window. Therefore, each of you will be arriving at randomly selected times between 3 p.m. and 4 p.m. Once each of you arrives at the mall, you will be there for exactly 15 minutes. When the 15 minutes are up, you leave.
- During the hour, there may or may not be an overlap between your and your friend’s visits. At some point, how many of you are present will reach a maximum number for the hour. This maximum could be one (sad!) or two. On average, what do you expect this maximum to be? The answer is between one and two.
- Hint: If you’re not sure where to start, think of the two arrival times on a coordinate plane. Your arrival time is the x-coordinate and your friend’s is the y. Which region in the coordinate plane are you considering in this puzzle? Which region results in the two of you meeting up?
- Instead of you and a friend, now suppose there are three total friends, yourself included. As before, you and the friends arrive at random times during the hour and each stay for 15 minutes.
- Hint: Again, at some point during the hour, there will be a maximum number of friends at the mall. This maximum could be one, two or three. On average, what would you expect this maximum number of friends to be?
- What about four total friends? On average, what would you expect the maximum number of friends meeting up to be?
- Hint: If you can’t find the exact answer, try finding an estimate. A computer might help.
- Suppose there are N friends. As N grows increasingly large, what would you expect the maximum number of friends meeting up to be, in terms of N?
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