Sorry if this is to simple; I have not done any Math in a while.
The scenario is , we have two participants/Runners: A,B both headed
randomly towards the same place P, and staying in P for a random interval
then heading out. We want to know the probability of
B arriving right after A leaves.
So we normalize meeting at $[0,1]$ and A arrives at $a_1$ and stays
until $a_2$; and similar pair $(b_1, b_2)$ for B; $0 \geq a_1,a_2,b_1,b_2 \leq 1 $.
So we want to compute the probability that $a_2=b_1$
Working on $[0,1] \times [0,1] $. The probability of the event $a_2=b_1$
is the measure of all rectangles with base $(a_1,a_2)$ and side $(a_2,1)$,
with total area $(a_2- a_1)(1-a_2)$. WE can compute the total area by the
$\int_0^1 (a_2-a_1)da_1 \int_0^1 (1-a_2)da_2 $
Getting 1/12 , which seems somewhat right intuitively. But I have been told
(non-rigorously) that the actual answer is 0. Can someone check and
criticque my work, please?