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 re(2): computer aided solution
I'm trying the 7-sided version.
Of course I've removed the rational exact-value calculation.
After evaluating a first-throw value of 1, the next thing it needs to evaluate is all the possibilities for 2,3,3,4,4. Evaluating just those possibilities takes 45 seconds. I'll see how far we get after an hour.
Posted by Charlie
on 2015-07-02 09:39:18