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

• Jan 11th 2012, 12:24 PM
divinelogos
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!
• Jan 11th 2012, 01:09 PM
emakarov
Re: Process A and Process B Each Start at Any Time Throughout the Day
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 $t_A$ and $t_B$ be the lengths of the processes, respectively, and let T be the day length. Then the processes overlap if $A-t_B\le B\le A+t_A$. This area is shaded in the following picture.

So, you need to find out the ratio of the shaded shape to the total area $T^2$.
• Jan 11th 2012, 02:10 PM
divinelogos
Re: Process A and Process B Each Start at Any Time Throughout the Day
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.
• Jan 11th 2012, 02:26 PM
emakarov
Re: Process A and Process B Each Start at Any Time Throughout the Day
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 $t_A$ and $t_B$ are much less than T, as in your example, then we can disregard the top and bottom trapezoids obtained by drawing horizontal lines $y = t_A$ and $y = T - t_B$. The area of the remaining middle parallelogram is $(T-t_A-t_B)(t_A+t_B)$. So, the probability is approximately $(T-t_A-t_B)(t_A+t_B)/T^2\approx (t_A+t_B)/T$.
• Jan 11th 2012, 03:00 PM
divinelogos
Re: Process A and Process B Each Start at Any Time Throughout the Day
So what would the final probability be? Thanks for your help.
• Jan 11th 2012, 03:08 PM
emakarov
Re: Process A and Process B Each Start at Any Time Throughout the Day
Do you need the final number or do you need a precise formula with no approximations? Why don't you make your own attempt?
• Jan 11th 2012, 04:43 PM
divinelogos
Re: Process A and Process B Each Start at Any Time Throughout the Day
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?