Dice & Probability (2D) — Sum Distributions, CLT & Casino Odds

Dice setup

Number of dice (n) 2

Sides per die (s)

Auto-roll speed 200/s

Rolls

Live readout

Theoretical mean

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Empirical mean

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Theoretical variance

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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 (s²−1)/12. By linearity the sum has E[Σ] = n·(s+1)/2 and Var[Σ] = n·(s²−1)/12. The exact distribution is the n-fold convolution of the uniform single-die distribution, computed here as polynomial coefficient counting, and it converges to a normal curve as n increases.

Casino games (two d6).

Bet	Win prob.	Payout	Expected value