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 One 1 to Six 6's (Posted on 2015-07-01)
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 No Rating

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 re(2): computer aided solution | Comment 7 of 14 |
(In reply to re: computer aided solution by Jer)

Concerning the ratio of the probability of a given number being the result, to that of the next number, hypothesized as sqrt(10) ~=  3.16227766016838, note the ratio of the probability of 1 to that of 2, as the number of sides progresses from 2 to 6 is:

3
3.166666666666667
3.217934165720772
3.233459959104922
3.238064338219506

getting farther from sqrt(10).

Ratio of p(2) to p(3) as m progresses from 3 to 6 follows a different trajectory:

3
3.069686411149826
3.092602637934237
3.099855399609466

 Posted by Charlie on 2015-07-02 15:34:07

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