I understand that if you wanted to compute the probability of rolling a specific three of a kind or more (getting three or more $1$s for example) then you would just calculate $$\sum_{k=0}^3(5^k){6\choose 6-k}\over 6^6$$ but I am not so sure how you would go about calculating the probability for any value to appear three or more times in a roll of six dice. Simply multiplying the total outcomes for obtaining three or more of a specific value is obviously not correct because you would count outcomes such as $(1,1,1,2,2,2)$ once for getting three $1$s and again for getting three $2$s.

Is there a general formula or method one can use to obtain the answer to not only the dice scenario, but any scenario where $P(X\ge x)$ for $X$ denoting the maximum amount of times any value appears in a string of $m$ values generated from $n$ different values?

Thanks in advance, and sorry if my wording is bad. I'm still not entirely sure of the best way to word the last part.

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    $\begingroup$ For the question in the title, you can bypass the issue you correctly noticed using inclusion-exclusion, giving $\dfrac{6\cdot (5^3\binom{6}{3}+5^2\binom{6}{2}+5^1\binom{6}{1}+1)-\binom{6}{2}\binom{6}{3}}{6^6}$ $\endgroup$ – JMoravitz Feb 10 '19 at 2:44
  • $\begingroup$ @JMoravitz Thanks for the reply and solution, I was just unsure of how I would count the terms that would be over counted so I could remove them. $\endgroup$ – S. Burc Feb 10 '19 at 17:38

There's more than one way. What's best depends on the relative sizes of $m$, $n$, and $x$.

First, we can go "forward" - enumerate all ways to have three or more of a kind, all ways to have three or more of two kinds, and so on, then run inclusion-exclusion. As noted in the comment by @JMoravitz, this works pretty well for your example of $x=3,m=n=6$.
For a particular $k$, there are $\binom{6}{3}\cdot (6-1)^3$ ways to have three of $k$, $\binom{6}{4}\cdot (6-1)^2$ ways to have four of $k$, $\binom{6}{5}\cdot (6-1)$ ways to have five of $k$, and $\binom{6}{6}$ ways to have six of $k$. Multiply by $6$ to get a total of $$6\left(\binom{6}{3}\cdot 5^3+\binom{6}{4}\cdot 5^2+\binom{6}{5}\cdot 5^1+\binom{6}{6}\right)$$ ways. But, of course, this is an overcount. We can have three each of two different kinds, in $\binom{6}{3}$ ways to choose which three are the "first" kind and $\binom{6}{2}$ pairs of kinds. These were counted twice by the preceding, so we subtract them off to get $$6\left(\binom{6}{3}\cdot 5^3+\binom{6}{4}\cdot 5^2+\binom{6}{5}\cdot 5^1+\binom{6}{6}\right)-\binom{6}{2}\cdot\binom{6}{3}$$ $$= 6(20\cdot 125+15\cdot 25+6\cdot 5+1)-15\cdot 20 = 17136$$ ways. Divide by $6^6=46656$ for the probability (about $0.367$).

Of course, as we increase $m$ for fixed $n$ and $x$, eventually we reach a point where that $x$ of a kind is certain (Pigeonhole principle). If we're close to that point, the forward count will be horrible - but there's an alternative. We can count the ways to avoid any instances of $x$ of a kind, and subtract the resulting probability from $1$. I'll demonstrate this backward count for the same example:

To avoid three of a kind, we must have between zero and 2 of each of the six kinds. I'll enumerate the possible ways to split $6$ into a sum of six terms each between zero and $2$: $2+2+2+0+0+0$, $2+2+1+1+0+0$, $2+1+1+1+1+0$, $1+1+1+1+1+1$.
For $2+2+2+0+0+0$, we have $\binom{6}{3}$ ways to choose which three numbers on the die show up, then $\binom{6}{2,2,2}=\binom{6}{2}\binom{4}{2}$ ways to choose which rolls each number gets.
For $2+2+1+1+0+0$, we have $\binom{6}{2,2,2}$ ways to choose which numbers are represented to each level, then $\binom{6}{2,2,1,1}$ ways to choose the rolls.
For $2+1+1+1+1+0$, we have $\binom{6}{1,4,1}$ ways to choose the numbers and $\binom{6}{2,1,1,1,1}$ ways to choose the rolls.
For $1+1+1+1+1+1$, we have one way to choose the numbers and $\binom{6}{1,1,1,1,1,1}=6!$ ways to choose the rolls.
Summing those up, that's $$20\cdot 90+90\cdot 180+30\cdot 360+1\cdot 720 = 29520$$ ways to avoid three of a kind. The probability we seek is $$1-\frac{29520}{46656}=\frac{17136}{46656}\approx 0.367$$ This backwards count never really goes too bad - the example here is a "worst" case of having the most possible ways to split $m$ - but it's particularly efficient when $m$ is close to $n(x-1)$, and it's instant when we're in the case $m > n(x-1)$ and at least one instance of $x$ of a kind is certain.

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  • $\begingroup$ Thank you very much for this detailed response! If I am understanding it correctly then I should be able to calculate the probability for $x=3, m=6$, and $n=124$ by doing $${124 \cdot \left({6\choose 3}\cdot 123^3+{6\choose 4}\cdot 123^2+{6\choose 5}\cdot 123^1+{6\choose 6}\right)-{124\choose 2}\cdot {6\choose3}}\over 124^6$$ Since there are $124\choose2$ pairs of kinds and $6\choose 3$ ways to arrange the first kind. I hope I am understanding the 'ways to arrange the first kind' correctly as different ways of ordering $(x,x,x,y,y,y)$. $\endgroup$ – S. Burc Feb 10 '19 at 18:39
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    $\begingroup$ Yes, that's correct. $\endgroup$ – jmerry Feb 10 '19 at 20:00
  • $\begingroup$ Okay, thank you very much. $\endgroup$ – S. Burc Feb 10 '19 at 20:08

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