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Two individuals agree to meet within the limits of the agreed period $\ l\ $. The first individual to arrive waits for the time $\ a <l$, and then leaves. What is the probability that they will meet?

I tried to use geometric probability.

let Ω = {$(x, y) | 0≤x, y≤ l$}
Using coordinate axes,

Let $(a_1) <(a_2)$ such that $(a_2) - (a_1) = a$ then the area where the coordinate pairs are found should be $[(a_1) ^ 2 - (a_2) ^ 2]$ and the area of Ω = $(l ^ 2)$

So the probability is P (A) = $([(a_1) ^ 2 - (a_2) ^ 2]) / (l ^ 2)$

coordinate axes

mielvil
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    I’m voting to close this question because: You are missing many key details from this classic homework question. Plus you should show your own attempts at, or at least ideas towards, solving it. – Graham Kemp Nov 06 '20 at 03:00
  • Welcome to MSE. You'll get a lot more help, and fewer votes to close, if you show that you have made a real effort to solve the problem yourself. What are your thoughts? What have you tried? How far did you get? Where are you stuck? This question is likely to be closed if you don't add more context. Please respond by editing the question body. Many people browsing questions will vote to close without reading the comments. – saulspatz Nov 06 '20 at 03:21
  • Related https://math.stackexchange.com/questions/103015/chance-of-meeting-in-a-bar – Henry Oct 14 '24 at 02:19

1 Answers1

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This is the correct drawing

enter image description here

The probability that they meet is the area of the purlple region on the total area, say 1 minus the area of the two white triangles on the total area

$$1-\frac{(l-a)^2}{l^2}$$

tommik
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