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?

 

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?

I can't follow what you are saying. What is Cx and Cy?

The only nonzero values of the joint prob function will occur for the following cases:

x=0, y=0
x=0, y=1
x=1, y=0
x=1, y=1
x=0, y=2
x=2, y=0

Note that x or y can be 0, 1 or 2 but x and y can not both be 2.

You should be able to calculate the probability of each case, noting that the sum of all probablities should add to 1.