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One 1 to Six 6's (Posted on 2015-07-01) Difficulty: 4 of 5
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.

No Solution Yet Submitted by Jer    
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m = 3 | Comment 1 of 14
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.

  Posted by Steve Herman on 2015-07-01 15:59:54
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