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.

Well, yes, that does look a little ugly.

When m = 3, the maximum number of rolls is 4.

3^4 = 81

By hand, I have counted up and

P(1) = 57/81 = 19/27

P(2) = 18/81 = 6/27

P(3) = 6/81 = 2/27

I am not even eager to tackle m = 4 by hand.

Waiting for inspiration to strike.