I’m trying to teach myself some probability. I’m solving some problems involving non-transitive dice. While I feel I understand most of the material, I don’t conceptually understand the following:

Suppose you have a single dice D with the following sides: 3 3 3 3 3 6

I’m trying to determine the probability of the sums when rolling two D dices.

Obviously, the only possible sums are: 6, 9, and 12.

It’s apparent to me that:

P(sum equals 6) = 5/6 * 5/6 = 25/36

P(sum equals 12) = 1/6 * 1/6 = 1/36

In my head, this is because (in the case of P(sum equals 6)), we have a 5/6 chance of rolling a 3 with the first die, and then we have a 5/6 chance of rolling a 3 with the second die.

However, when I apply this logic to P(sum equals 9), it doesn’t work, since my logic dictates that we have 5/6 chance of rolling 3 with the first die, and then 1/6 chance of rolling 6 with the second die.

But this doesn’t make sense, since P(sum equals 9) = 1 – P(sum equals 6) + P(sum equals 12)

So, I can only think that I’m thinking about this in a fundamentally incorrect way.

NOTE:

I can see that we can say P(sum equals 9) = 5/36 + 5/36, but I don’t understand why we would sum these? Doesn’t this mean that we are saying something like:

“the first die could be 3 and then the second die could be 6 OR the first die could be 6 and the second die could be 3”

Isn’t that wrong because it’s not relevant which die is rolled first? Why would the number of possible rolls double? Don’t we simply choose an arbitrary die and roll it? It seems like this is the inclusion-exclusion principle at work, but I don’t know how to put it in words that I can intuitively understand. Right now, it feels contrived, like I’m just guessing around and blindly following a formula.