How do i derive the joint probability distribution table of X and Y? If there is a bag with 3 red balls, 2 blue balls and 1 white ball. Two balls are drawn without replacement. Let X be number of red balls drawn and Y be number of blue balls drawn.
 A: Step 1: Determine the sample space.
When you draw a sample of two balls from this bag

all of the 15 following results are equally likely:

To assure that this figure is complete and contains no redundancies, the samples are drawn systematically, beginning with all those containing ball $A$, then all additional samples containing $B$ (but not $A$), then all additional samples containing $C$ (but not $A$ or $B$), etc.
Step 2: Label the elements of the sample space with the values of the random variables.
Next to each sample, by counting the number of red balls and the number of blue balls, I have drawn the values of $X$ and $Y$ in the form $X,Y$.
Step 3: Summarize the probabilities.
We can total the probabilities of each possible value of $(X,Y)$ and present these totals in a table.  For instance, two of the samples have $(X,Y) = (0,1)$, giving a probability of $2/15$ for this value of the (bivariate) random variable $(X,Y)$, which is entered in the column $X=0$ and the row $Y=1$.
$$\begin{array}{cccc}
  & X=0 & X=1 & X=2 \\
 Y=2 & \frac{1}{15} & 0 & 0 \\
 Y=1 & \frac{2}{15} & \frac{6}{15} & 0 \\
 Y=0 & 0 & \frac{3}{15} & \frac{3}{15} \\
\end{array}$$
A: To find the joint distribution of X and Y, think about all the possible values that X and Y can take.  Then, the joint distribution is just the probability that X and Y simultaneously take two of their possible values.
