# Probability of a dice being in a set of dice

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)

• To be honest I'd want to say that because P1 is making this claim, he has at least one $5$ – John Fernley Jun 27 '14 at 3:05
• I agree, I think it is a good assumption – sontek Jun 27 '14 at 5:25
• Did you intend P1 to throw four rather than five dice? – Henry Jun 27 '14 at 5:39

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.
• $$\binom{9}{2}=\frac{9!}{2!7!}=36$$ – John Fernley Jun 27 '14 at 3:00
• oh, and $\binom{n}{k}$ is always the number of ways to choose $k$ objects from $n$ objects – John Fernley Jun 27 '14 at 3:02
• The problem is that P1's statement reveals something about the unknown number of $5$s that P1 has, so you actually need a conditional probability, possibly using game theory which would depend on rules of the game we have not seen such as who wins. – Henry Jun 27 '14 at 5:39