Yakk

2006-12-23, 03:41 PM

E is "Expected value". Otherwise known as "average".

V is "Variance". It measure how spread out your results are.

1.96*sqrt(V) is the statistical 95% confidence interval. In general, such a confidence interval only really works after you add up a reasonable number of random events (such as 3d6).

E(1dX) = (1+X)/2

V(1dX) = (1+X)(1-X)/12

V(1d1) = 0 (aka, +1)

V(1d2) = 3/12 = 1/4

V(1d3) = 8/12 = 2/3

V(1d4) = 15/12 = 1 & 1/4

V(1d6) = 35/12 =~ 2.9

V(1d8) = 63/12 = 5 & 1/4

V(1d10) = 99/12 = 8 & 1/4

V(1d12) = 143/12 =~ 11.9

V(1d20) = 399/12 = 33 & 1/4

V(1d100) = 9999/12 = 833 & 1/4

Both E and V are linear when you add multiple dice together. Ie:

V(2d6) = V(1d6)*2

and V(1d4+1d6-1d8) = V(1d4) + V(1d6) + V(1d8)

When you multiply dice by a constant K, E goes up by K, while V goes up by K^2.

Ie:

V(1d6 * 2) = 4 * V(1d6) > V(2d6)

E(1d6 * 2) = 2 * E(1d6) = E(2d6)

If you have a situation like "roll 1d20. On a 20, event A happens, otherwise event B happens"

E = 1/20 * E(A) + 19/20 * E(B)

V = 1/20 * V(A) + 19/20 * V(B) - 2 E(A)*E(B)*1/20 * 19/20

If the chances are different, the 1/20 and 19/20 fractions change.

So, for hit-or-miss, where B is miss, E(B) = 0 and V(B) = 0. So

E = hit_chance * E(hit)

V = hit_chance * V(hit)

For crits... well it gets more complicated. :)

V is "Variance". It measure how spread out your results are.

1.96*sqrt(V) is the statistical 95% confidence interval. In general, such a confidence interval only really works after you add up a reasonable number of random events (such as 3d6).

E(1dX) = (1+X)/2

V(1dX) = (1+X)(1-X)/12

V(1d1) = 0 (aka, +1)

V(1d2) = 3/12 = 1/4

V(1d3) = 8/12 = 2/3

V(1d4) = 15/12 = 1 & 1/4

V(1d6) = 35/12 =~ 2.9

V(1d8) = 63/12 = 5 & 1/4

V(1d10) = 99/12 = 8 & 1/4

V(1d12) = 143/12 =~ 11.9

V(1d20) = 399/12 = 33 & 1/4

V(1d100) = 9999/12 = 833 & 1/4

Both E and V are linear when you add multiple dice together. Ie:

V(2d6) = V(1d6)*2

and V(1d4+1d6-1d8) = V(1d4) + V(1d6) + V(1d8)

When you multiply dice by a constant K, E goes up by K, while V goes up by K^2.

Ie:

V(1d6 * 2) = 4 * V(1d6) > V(2d6)

E(1d6 * 2) = 2 * E(1d6) = E(2d6)

If you have a situation like "roll 1d20. On a 20, event A happens, otherwise event B happens"

E = 1/20 * E(A) + 19/20 * E(B)

V = 1/20 * V(A) + 19/20 * V(B) - 2 E(A)*E(B)*1/20 * 19/20

If the chances are different, the 1/20 and 19/20 fractions change.

So, for hit-or-miss, where B is miss, E(B) = 0 and V(B) = 0. So

E = hit_chance * E(hit)

V = hit_chance * V(hit)

For crits... well it gets more complicated. :)