Binomial Distribution on Dice with Multiple Values

I asked a question about some dice probability a while back for a hobby of mine, and was greatly helped by an introduction to Binomial Distribution (https://math.stackexchange.com/a/2355623?noredirect=1). However, I'm hoping to change up my system a little bit.

So I'm trying to find the probability of getting Y successes using X dice, where the probablilty of a success is 1/3 (a 4 or a 5); I already have equations set up for this. What I'd also like to find out is how having a 6 be worth 2 successes would work out (along with 4 and 5 being worth 1 success). Ideally I'd like to know if there's any way to incorporate this into excel where the majority of my data is already.

To sum up: Proability of getting Y in X trials, with a 1/3 chance of getting 1 success, and a 1/6 chance of 2 successes.

• Let $(Z_1, Z_2, Z_3)$ be the number of $6$, $4$ or $5$, and $1$ or $2$ or $3$ respectively in $x$ trials. Then they jointly follows $\text{Multinomial}(x; 1/6, 1/3, 1/2)$. Note the number of successes $Y = 2Z_1 + Z_2$. The pmf of $Y$ can be worked out for certain, but I just doubt that if there is a nice pattern which allow you to avoid the recursive process in finding the pmf. – BGM May 11 '18 at 17:05
• Hmm, so I can't say that I'm all that mathematically inclined, so I don't fully understand your answer. I see where you got the factions for the Multinomial of course, but I don't really see how it fits together. – Telethia May 12 '18 at 5:27