Geometric Probability in Multiple Dimensions

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?

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?