Geometric Probability in Multiple Dimensions

Question

I understand how Geometric Probability works in $1$, $2$, and $3$ dimensions, but is it possible to do these problems in, say, $5$ dimensions? For example,

Five friends are to show up at a party from $1:00$ to $2:00$ and are to stay for $6$ minutes each. What is the probability that there exists a point in time such that all of the $5$ friends meet?

All I have is that I can let the number of hours after $1:00$ where each of the friends reach the party at be $a$, $b$, $c$, $d$, and $e$. WLOG $0\le a \le b \le c \le d \le e \le 1$. Now I am confused on how to make a diagram to solve the problem.

Note: There is a similar discussion on this, but the explanations given use calculus, which I don't understand at all. Can somebody please provide an elementary solution? Thanks!

Answer 1

Please see if this helps:

Consider the simpler problem of 2 people. Chart on the x axis the time of stay of first person. Chart on the y axis the time of stay of second person. The cartesian product of these intervals together form a rectangle in $\mathbb R^2$

Question: When do the two meet, and what does this mean geometrically? It's easy to see that whenever this rectangle intersects the $x=y$ line, then the two people meet (or have an overlapping time), or else if the rectangle is fully below $x=y$, or fully above they don't meet.

Extending to higher dimensions: When formulated this way, it should be the case that in $\mathbb R^n$, when the rectangle formed by the cartesian product of the intervals of stay of different people intersects the line $x_1 = x_2 = \dots = x_n$, (or in parametric form, $\lambda[1,1,\dots,1]\quad\forall \lambda$) then, all the people have some overlapping time of stay.

Probability of overlap: The probability is now the measure of the union of all the rectangles (or hypercubes in $\mathbb R^n$) intersecting with the line ($x_1=x_2=\dots$) vs the measure of the entire space under consideration, say under normalization, $[0,1]\times[0,1]\times\dots[0,1]$