# Roll an n-sided die m times. What is the probability that (1, 2, 3 … n) faces have appeared?

If I roll an n-sided die m times, what is the probability of getting only one face, two faces, .... or n faces?

Let us assume that I have a 6-sided die and I roll it 10 times. I want to calculate different probabilities. The probability of getting only one face from all rolls. The probability of getting only 2 faces in all rolls and so on until calculating the probability of the appearance of the 6 faces (at least one time for each face) in all rolls.

• Could you clarify. – marshal craft Nov 1 '17 at 6:37
• OK, can you calculate the said probabilities after one roll? How about two rolls? – Ivan Neretin Nov 1 '17 at 7:32

The probability that $x$ faces have appeared at least once after $m$ throws of a fair $n$-faced die is $$Pr(X=x \mid m) = \frac{S_2(m,x) \, n!} {(n-x)!\,n^m}$$ where $S_2(m,x)$ is a Stirling number of the second kind

So for example with $n=6$ and $m=10$ you would get $$Pr(X=2) = \frac{511 \times 720} {24 \times 60466176} \approx 0.000254$$

You could calculate this recursively using $$Pr(X=x \mid m)=\frac{n-x+1}{n}Pr(X=x-1 \mid m-1) + \frac{x}{n}Pr(X=x \mid m-1)$$ starting with $Pr(X=0 \mid m)=0$ and $Pr(X=x \mid 0)$ except $Pr(X=0 \mid 0)=1$. That would give a table like this:

    x   1           2           3           4           5           6
m
1       1           0           0           0           0           0
2       0.166667    0.833333    0           0           0           0
3       0.027778    0.416667    0.555556    0           0           0
4       0.004630    0.162037    0.555556    0.277778    0           0
5       0.000772    0.057870    0.385802    0.462963    0.092593    0
6       0.000129    0.019933    0.231481    0.501543    0.231481    0.015432
7       0.000021    0.006752    0.129029    0.450103    0.360082    0.054012
8       0.000004    0.002268    0.069016    0.364583    0.450103    0.114026
9       0.000001    0.000759    0.036020    0.277563    0.496614    0.189043
10      0.0000001   0.000254    0.018516    0.203052    0.506366    0.271812

• Many thanks dear, this is really helpful. – user498036 Nov 1 '17 at 23:36