Question

Romeo and Juliet have a date at a given time, and each will arrive at the meeting place with a delay between 0 and 1 hour, with all pairs of delay being equally likely. The first to arrive will wait for 15 minutes and will leave if the other has not yet arrived. What is the probability that they will meet?

Answer 1

So let x be the time when Romeo arrives.

He can arrive at any time between 0 and 1,

let y be the time when Julia comes.

So the points [x,y] of the square are every possible combinations of times when they comes, if I can say it like that.

The area of the square if 1.

The situation that they come at the same moment is symbolized by the diagonal.

But one of them can arrive even 15 minutes later - the upper line symbolize the situation when Romeo will wait for Julia 15 minutes exactly, so the area between diagonal and upper line symbolize the situation when Romeo will wait for Julia 15 minutes or less.

The lower line, on the other hand, represents the situation when Julia will wait for Romeo 15 minutes exactly, so the area between this line and the diagonals means that Julia is waiting for Romeo 15 minutes or less.

Together, these two areas gives all possible "time-combinations" at which they'll meet.

So now the probability

P(they meet) = the area of the part in which they meet / the area of the square

                        =        7 / 16 =0.4375

OR

= 1- area of two unshaded triangles

= 1 - ( 2*(3/4)*(3/4)) / 2

= 1- (1.125/2)

= 1- 0.5625

=0.4375