š² Dice & Probability
Roll n dice with s sides. The exact distribution of the sum is found by convolution; as n grows it approaches a Gaussian (the central limit theorem), and the empirical mean and variance converge to the theoretical values (the law of large numbers).
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Theoretical mean
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Empirical mean
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Theoretical variance
ā
Empirical variance
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Most likely sum
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Total rolls
The maths. For n dice of s sides,
each die has mean (s+1)/2 and variance (sĀ²ā1)/12. By linearity the sum
has E[Ī£] = nĀ·(s+1)/2 and
Var[Ī£] = nĀ·(sĀ²ā1)/12. The exact distribution is the
n-fold convolution of the uniform single-die distribution ā
computed here as polynomial coefficient counting ā and converges to a
normal curve as n increases.
Casino games (two d6).
| Bet | Win prob. | Payout | Expected value |
|---|