Three Balls in Three Numbered Cells

Question

The question asks:

Three balls are randomly thrown into three numbered cells.

Let X be the number of cells that contain at least one ball.

Let Y be the number of balls in cell 1.

What is the joint probability mass function of X and Y?

My attempt: I understood the question as the three balls being the same, and this is the joint probability mass function (along with 10 possible combinations) I got. Is this correct?

Is there a difference whether or not each individual ball is different or if all three balls are the same?

Because I was marked wrong for the joint probability mass function during the exam, and I don't know whether it was my math that was wrong of if I interpreted the question wrongly?

Answer 1

You assume that all 10 possible outcomes are equally likely, but that is not the case.

For example, to get all 3 balls in bin 1 has a probability of $\frac{1}{27}$, but to get 1 ball in each of the bins has a probability of $\frac{6}{27}$. And to get 2 balls in one bin and the third in a different bin has a probability of $\frac{3}{27}$

Answer 2

The PMF is the same in both cases, because you are not sampling. If you want to compute the conditional probability of one related event given that you observed one ball and where it is, then it is relevant and will change your probabilities. But for the mass function is irrelevant.