# Simple Dice Rolling Problem

If you play poker dice by simultaneously rolling 5 dice, why is $P\text{{five alike}} =.0008$?

I guess I understand the fact that each dice has the probability to land on the same number $1/6$ of the time making it $6\cdot (1/6)^5$, but I don't understand why this alternate approach does not work:

The question can be seen in the form that there are $n=5$ indistinguishable objects lined up and we want to divide them into $r=6$ groups.

There are $\pmatrix{n+r-1\cr r-1\cr }$ distinct nonnegative integer valued vectors $x_1, x_2, ...$ satisfying $x_1+x_2+...x_r=n$, thus for the particular question there would be a total of $\pmatrix{10\cr5}$ different combinations of dice you can roll, with 6 combinations of dice that are five alike such that the probability of getting five alike is $6/252$.

Thank you very much for any help, and excuse my poor formatting. This is my first time using the Math Tex thing.

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## 1 Answer

Roll the dice one at a time. Consider the result of the first roll. The probability that the next $4$ match it is $\left(\frac{1}{6}\right)^4$. This is approximately $0.0007716$.

As to the question about why the alternate approach does not work, you have produced a sample space of size $252$. However, not all members of this sample space equally likely.

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Thank you very much for the response. As a follow up question, I used this method because a semi-well known Youtube channel that specializes in posting math puzzles did something similar to this. His original problem is here, starting at 1:00 :youtube.com/watch?v=ILAGckNjXvo. And his solution is here at 2:15 : youtube.com/watch?v=E31aGpE8TH0. When he says that the "chances are 1/120" is he wrong or is his problem inherently different than mine? –  Matt Apr 19 '12 at 19:33
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