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.
2-sided die:
Probabilities:
{0.75,0.25}
Ratios:
{3.}
3-sided die:
Probabilities:
{0.703704,0.222222,0.0740741}
Ratios:
{3.16667,3.}
4-sided die:
Probabilities:
{0.692139,0.215088,0.0700684,0.0227051}
Ratios:
{3.21793,3.06969,3.08602}
5-sided die:
Probabilities:
{0.689011,0.213088,0.0689024,0.0220867,0.00691251}
Ratios:
{3.23346,3.0926,3.11964,3.19517}
6-sided die:
Probabilities:
{0.688152,0.212519,0.0685579,0.0218963,0.00681664,0.00205811}
Ratios:
{3.23806,3.09986,3.13102,3.21219,3.31208}
7-sided die:
Probabilities:
{0.687918,0.21236,0.0684577,0.0218387,0.0067864,0.0020436,0.000595671}
Ratios:
{3.23939,3.10206,3.1347,3.21801,3.32081,3.43075}
8-sided die:
Probabilities:
{0.687855,0.212316,0.0684294,0.0218218,0.00677717,0.00203898,0.000593553,0.000167281}
Ratios:
{3.23977,3.10271,3.13583,3.2199,3.3238,3.43521,3.54825}
9-sided die:
Probabilities:
{0.687839,0.212305,0.0684216,0.021817,0.00677445,0.00203757,0.000592881,0.000166985,0.000045586}
Ratios:
{3.23987,3.10289,3.13616,3.22048,3.32477,3.43673,3.55051,3.66307}
10-sided die:
Probabilities:
{0.687835,0.212302,0.0684195,0.0218157,0.00677368,0.00203716,0.000592677,0.000166892,0.0000455465,0.000012067}
Ratios:
{3.23989,3.10294,3.13625,3.22065,3.32506,3.43722,3.55127,3.6642,3.77447}
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Posted by Daniel
on 2015-07-03 21:01:10 |
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