A standard six-sided die is to be rolled repeatedly until a side appears a number of times equal to its number. In other words until the n-th n appears.

Let P(n)=the probability the game terminates with the n-th n.

Find the distribution of n.

Feel free to generalize for m sides.

Warning: I have not managed this past m=4.

(In reply to computer aided solution by Charlie)

I was worried this might be a little on the hard side.  I was also holding out hope that I was missing something...

There is a glimmer of structure here:
The probabilities go down be a factor of very nearly √10.
On closer inspection so do my m=4 solutions.

If you feel like calculating the distribution for other values of m I'd like to investigate further.  I have enough to know OEIS doesn't have related sequences so far.

Posted by Jer on 2015-07-01 22:17:03