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.