Probability of meeting someone between two times of day, within a predefined time?

[moonman239]

Two people agree to meet between 2 and 4 pm, with the understanding that each will wait no longer than 15 minutes for the other. What is the probability that they will meet? (This is NOT homework) The book I borrowed has the equation but I don't have it in front of me right now. Plus, the equation seems complicated.

 

[moonman239]

Here's my amateur approach to a solution: P(not arriving within 15 minutes of each other) = (8-2+1)/8

 

[Redbelly98]

Moderator's note: thread moved from Set Theory, Logic, Probability, Statistics.

Our policy on "Homework Help" applies to any textbook-style problem, whether it's for an actual course or just independent study.

 

[Tedjn]

Think geometrically. Let the x-axis from 2 to 4 indicate when person 1 might arrive, and let the y-axis from 2 to 4 indicate when person 2 might arrive. Any point in that rectangle is a possibility. What is the area in which they successfully meet?

 

[moonman239]

What is the area in which they successfully meet?

What do you mean by that?

 

[moonman239]

Never mind, I found it.

 

[moonman239]

Could you please explain the solution in pre-algebraic terms?

 

[Tedjn]

Think about it in this way, the two people can come at any time between 2 to 4. We can write these two times down as an ordered pair (x,y), with 2 ≤ x,y ≤ 4. The first coordinate is the time at which the first person arrives; the second coordinate is the time at which the second person arrives. Any such point in that region (call it R) is equally likely. Here, we are treating time as completely continuous, which is not a bad approximation.

Now, find the ordered pairs inside this region for which the coordinates differ by less than 15 minutes (i.e. 0.25 hours). (Hint: inequalities). The two people will meet in this region (call it A). You are looking for the probability that, upon throwing a dart at R, the dart lands in A. Convince yourself that this is given by Area(A)/Area(R).

 

[LCKurtz]

For an old thread discussing a similar problem see

https://www.physicsforums.com/showthread.php?t=397292