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Math Help - Process A and Process B Each Start at Any Time Throughout the Day

  1. #1
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    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!
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  2. #2
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    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.
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  3. #3
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    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.
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    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.
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    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.
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  6. #6
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    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?
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  7. #7
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    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?
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