This isn't a homework problem just a simplified version of a situation from a video game that has been bothering me that I couldn't figure out after a couple hours of googling and thinking.

Imagine a fair three sided dice that is labeled with A,B and C. I chose letters cause I thought the numbers would confuse me. You roll this dice 9 times. What are the chances that you roll 3 or more A's?

I tried solving the problem for when you roll fewer times so I could work my way up and I'm not sure if got those examples right. If you roll 3 times the answer is obviously 1/27.

For 4 times I got an answer of 9/81 and I'm fairly sure of that. And for the case of 5 rolls I got 31/243. The probability of getting 3 or more A's only increased around one percentage point so I'm not sure of this result either.

So I tried coding something to just make a list of all possible permutations of this scenario and count the number of elements in the list fit the criteria but I couldn't even make the list. I could just do the case of rolling the dice 5 times by hand but that doesn't help me answer the actual question.

So if anyone that is good at math or programming could help me out that would be great.