Thread: Process A and Process B Each Start at Any Time Throughout the Day

This isn't a homework question, but rather a question of personal interest:

Some process A starts at a random time during the day and runs for 30 seconds. Process B also starts at a random time during the day and runs for
6 minutes. What is the probability that for a given day, the
processes overlap?


Does this problem involve independent random variables and/or calculus, since time is continuous and we consider an interval with infinitely many possible starting points? I was thinking the distance between the starting points could be viewed as a differential, and some type of integral would be involved in computing the probability. Any help would be appreciated! Thanks!

Let A be a random variable denoting the start time of process A, and let B be the start time of process B. Also, let and be the lengths of the processes, respectively, and let T be the day length. Then the processes overlap if . This area is shaded in the following picture.



So, you need to find out the ratio of the shaded shape to the total area .

Any ideas on how to compute that ratio? I wonder if my initial thought about the integral was correct, given that we're dealing with areas.

In general, integrals may be necessary, but when A and B are distributed uniformly, it's easy to break the shape into triangles, rectangles, trapezoids and parallelograms. If and are much less than T, as in your example, then we can disregard the top and bottom trapezoids obtained by drawing horizontal lines and . The area of the remaining middle parallelogram is . So, the probability is approximately .

So what would the final probability be? Thanks for your help.

Do you need the final number or do you need a precise formula with no approximations? Why don't you make your own attempt?

I could come up with an approximation, but I'd like to see the correct way of doing it. It seems like this is a problem that could be solved by a standard method in probability theory. Do you know what that standard method would be?