MHB Solving Probability Coupling Problems: X & Y

[Harambe1]

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.

 

[Prove It]

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?

 

[HallsofIvy]

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.