X and Y are two dependant random variables. Find the distribution of C if C = Y-X

Question

There are 20 balls in a bag and 7 of them are red. If you pick a red ball, you take it out from the bag. Otherwise, you put it back in.

Let X: The number of red balls you pick after 2 tries

Let Y: The number of red balls you pick after 3 tries

Question: Let C = Y-X, calculate the distribution of C.

I start by finding the distribution of X and Y with the help of a tree diagram:

Then, I believe I need to make a joint distribution of X and Y: It took me a while to figure it, but here it is:

I know that X and Y are mutually exclusive for sure too. But now I'm stuck because I don't know what to do with it. I have absolutely no idea how to find the distribution of C.

I need to fill the distribution for C, if C = Y-X:

Any help is appreciated.. Thank you.

Answer 1

(Assuming your joint distribution is correct ...) Each cell in the table of the joint distribution gives a value of $p(x,y)=P(X=x,Y=y).$ Now $C=Y-X$, so for each $c$, $p(c)=P(C=c)=P(Y-X=c)$ is just the sum of all those cells that have $y-x=c:$