What is the difference between these two distributions?

Question

I am working on the OCW assignments on information theory and in the 1st question, I am getting confused between the total number of sequences that Bob will eat the cookies. Let's say Bob takes out 2 cookies (X and Y) sequentially and eats them. So the joint distribution matrix will be something like this:

X\Y    Pb      C         O
Pb  3/6*2/5  3/6*2/5  3/6*1/5
C   2/6*3/5  2/6*1/5  2/6*1/5
O   1/6*3/5  1/6*2/5  0

However, the total combinations available to us in which Bob can eat is quite different, which are:

X   Y
Pb Pb
Pb Ch
Pb O
Ch Pb
Ch Ch
Ch O
O Pb
O Ch

So the probability distribution is different here. So why is this happening?

Answer 1

There is nothing mysterious going on. There is 0 probability of eating the sequence O O so it hasn't appeared in your "total combinations" list. That length is list 8 while the matrix has 9 entries. Ch in the list stands for C in the matrix.

And surely you know that the row "X" in the matrix corresponds to the first cookie that Bob eats (the first column in the list) and the column "Y" in the matrix corresponds to the second cookie that Bob eats (the second column in the list).

Each row in the total combinations list has probability of happening equal to the entry in the matrix corresponding to it. This is a discrete distribution.