How to solve two balls select event without replacement

Question

Q)A box contains 4 red balls, 3 green balls and 3 blue balls. Two balls are selected at random without replacement. Let X represent the number of red balls in the sample and Y the number of green balls in the sample. a) Arrange the different pairs of values of (X, Y ) as the cells in a table, each cell being filled with the probability of that pair of values occurring, i.e. provide the joint probability distribution. b) What does the random variable Z = 2 - X - Y represent? c) Calculate Cov(X, Y ). d) Calculate P(X = 1 | -2 < X - Y < 2).

I couldn't understand how to think to solve the part a) in this question and so on.

Answer 1

To solving this question first of all you have to create a tree with this two events. First data in this question is that we can take is , these are not independent event. so you can create tree like this,

In first part you have to create the joint table of X and Y. there is 0,1,2 are the only possible values that each variable can get. The critical situations are that X-1 , Y-0 and X-0 , Y-1 .Because they got two possible chances in same situation that one color ball take first and that same color ball select the second time. So this is the table that can get according to this tree.

part b represent the blue balls in selected sample