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  1. MHB Apprentice

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    #1
    Hi, I'm struggling to understand probability coupling. I have the following problem:

    Let X and Y each be uniformly distributed on the discrete set {1,...6} (i.e. the distribution of the roll of 1 fair die).
    (a) If X and Y are independent, what is Pr[X = Y]?
    (b) Couple X and Y so that Pr[X = Y] = 1.
    (c) Couple X and Y so that Pr[X > Y] = 5/6.

    I'm not entirely sure where to start and can't find much information on it.

    For part (b), would I be right in simply saying "Let X={1,2,...,6} and let Y={X}. Thus, Pr[X = Y] = 1." If so, great but if not where am I going wrong?

    Thanks for your help.
    Last edited by Harambe; October 31st, 2016 at 17:09.

  2. MHB Master
    MHB Math Helper

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    #2
    Quote Originally Posted by Harambe View Post
    Hi, I'm struggling to understand probability coupling. I have the following problem:

    Let X and Y each be uniformly distributed on the discrete set {1,...6} (i.e. the distribution of the roll of 1 fair die).
    (a) If X and Y are independent, what is Pr[X = Y]?
    (b) Couple X and Y so that Pr[X = Y] = 1.
    (c) Couple X and Y so that Pr[X > Y] = 5/6.

    I'm not entirely sure where to start and can't find much information on it.

    For part (b), would I be right in simply saying "Let X={1,2,...,6} and let Y={X}. Thus, Pr[X = Y] = 1." If so, great but if not where am I going wrong?

    Thanks for your help.
    Start by drawing up a 2-way table which shows what you can roll from the two dice. How many possibilities are there? How many of those possibilities have the rolls the same?

  3. MHB Journeyman
    MHB Math Helper

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    #3
    There are 6 possible outcomes for each of X and Y so there are 36 possible outcomes for (X, Y). In 6 of those X= Y.

    For b, you say "let Y= {x}". What are the braces intended to mean here? Why not just "Y= X"?

    For the third, write out all 36 possible outcomes. Find a subset containing 6 of those outcomes such that X> Y in 5 of them. It is not necessary that you be able to write a "formula" describing them.
    Last edited by HallsofIvy; December 29th, 2016 at 09:06.

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