I have no formal background in math, statistics, or anything. Just trying to figure out a fun problem with a game of dice.
Lets say you have 3 people sitting a table rolling dice (including yourself).
This is what the result set looks like:
You: [5, 4, 3, 2, 1] P1: [x, x, x, x] P2: [x, x, x, x, x]
So you know what dice you have but not what the others have. Now P1 guesses that there are 3 fives on the table. What is the probability that there are actually 3 fives on the table?
Since there are 9 unknown dice, I know there are 531,441 possibilities that could have been rolled (9**6). But past that I'm not sure what to calculate next.
EDIT: After thinking about it, 9**6 is probably invalid because although someone could get [5, 4, 3, 2, 1] and [1, 2, 3, 4, 5] that is the same thing for what I'm trying to do, which is find out how probable it is to have X amount of a specific face value (i.e 4 fours)
You need exactly $2$ of the remaining $9$ dice to be $5$'s for P1 to be correct.
There are $\dbinom{9}{2}$ ways to pick the $2$ other dice which need to be $5$'s.
The other $7$ dice need to show one of $\{1,2,3,4,6\}$, so there are $5^7$ ways for this to happen.
Now, you should be able to calculate the total number of ways for exactly $2$ of the remaining $9$ dice to be $5$'s, and thus, the probability that P1 is correct.