- #1

Tim67

- 6

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Last night, Dave and Kathy both arrived at Pizza Palace at two different random times between 10:00 p.m. and midnight. They had agreed to wait exactly 15 minutes for each other to arrive before leaving. What is the probability that Dave and Kathy were together at Pizza Palace last night between 10:00 p.m. and midnight?

I saw this on a blog, with the solution, so I'm not really asking what the answer is or how to get there, but I'm just wondering what's wrong with my reasoning, which gets me close to, but not at, the right answer:

Think of a number line: 1, 2, 3, ... 120, representing all minutes either can enter the place. So for any minute A enters, B can enter at any of 120 different minutes, so the total combinations of 'ways' both can arrive throughout the night is 120^2.

So, I think of three cases: A arrives in first 15, last 15, or any of the other 90.

Arriving in any of the middle 90 minute marks: B can arrive at any of the 15 minute marks within A's 15-minute interval, or can arrive any of 14 minutes before A arrives and they will still meet each other. So for each 90 minute marks we have 15+14 opportunities, so 90*29.

Either first 15 or last 15: A arrives at minute 1, B has 15 minutes to arrive; A arrives at minute 2, B has A's 15 minute interval or 1 back = 16 opportunities, etc. Same reasoning for last 15 minutes. So: 2(15 + 16 + 17 + ... 29)

so [2(15+..+29) + 90*29]/120^2 = .227

the right answer is .234