What is the Joint Probability Function for Rolling a Balanced Die Twice?

[kasse]

Consider an experiment that consists of 2 rolls of a balanced die. If X is the number of 4s and Y is the number of 5s obtained in the 2 rolls of the die, find the joint probability function.


$$f(x,y) = \frac{(_{2}C_{x})(_{2}C_{y})}{36}$$

because there are $$_{2}C_{x}$$ ways to combine x 4s and $$_{2}C_{y}$$ ways to combine y 5s, and 36 possible combinations from a throw with 2 dice.

However, this formula doesn't give me the correct probability distribution of X and Y. What's my mistake?

 

[HallsofIvy]

The number of 5s and 4s are not independent. If you roll two 5s you MUST roll 0 4s. If you roll 1 5 you MUST roll 1 4. If you roll 2 4s you MUST roll two 4s.
The correct probability of rolling x 5s and y 4s is exactly equal to the probabilty of rolling x 5s (x 0, 1, or 2) which is also equal to the probability of rolling y= 2-x 4s.