What you are talking about is probability. Yet nobody is using math.

*sigh*

So, a simple way to break down a given random number generator is mean and standard deviation. If you then do a CDF (cumulative distribution function) of the results, translated based on mean and scaled based on standard deviation, the graphs look quite similar. :) The differences mainly occur at the far end (at the < 5% of events scale).

The standard deviation of a 1dX is sqrt((X^2-1)/12).

The standard deviation of YdX is sqrt(Y * (X^2-1)/12 )

So we have:

E(1d20) = 10.5

E(2d10) = 11

SD(1d20) =~ 5.77

SD(2d10) =~ 4.06

The lower SD on 2d10 is an expression of the "tighter" distribution.

The rough conversion between (1d20+N) to (2d10+M) is:

(N - 10.5) = 1.4*(M-11)

(1.4 is 5.77/4.06).

Do you want modifiers to the d20 and difficulty to be roughly 1.4 times as important? Because you did that, regardless of your goal one way or other.

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So that describes what happens most of the time. The next problem is dealing with the "tails".

The critical distribution above:

is very different than the standard D&D criticals.20/x2 -> 20/x3.5 (calculate as if x4, but deal only half damage on the final ‘hit’)

19-20/x2 -> 19-20/x3

18-20/x2-> 18-20/x2

20/x3 -> 20/x5

20/x4 -> 20/x6

P(2d10 >= 20) = 0.01

P(2d10 >= 19) = 0.03

P(2d10 >= 18) = 0.06

P(2d10 >= 17) = 0.10

P(2d10 >= 16) = 0.15

P(2d10 >= 15) = 0.21

P(2d10 >= 14) = 0.28

P(2d10 >= 13) = 0.36

P(2d10 >= 12) = 0.45

For (21-C) >= 12, we get:

P(2d10 >= 21-C) = (C+1)*(C)/200

In comparison, the P(1d20 >= (21-C)) = C/20

The ratio works out to be:

P(1d20 >= (21-C)) / P(2d10 >= (21-C)) = 10/(C+1)

That is the ratio of 1d20 crit chance to 2d10 crit chance. Notice how it is rather large -- at C=1 (ie, 20s only), you need crits to be 5 times larger to be just as good!

However, we need to balance both keen/imp crit and normal. Given the shape of the curve, I'd recommend that keen/imp crit should do something slightly different.

I'd vote for "increase crit multiplier by A, and increase crit width by B".

Using that, we can now figure out what has to happen to the base crit multiplier for keen weapons to be about as good and base weapons to be about as good, on the average. (they will probably have a worse variance).

Also, should we change the baseline crits? I'm thinking "yes".

Because 20x2 under d20 has to become 20x6 under 2d10 to be just as good. And 20x4? 20x16! That's getting silly.

First proposal:

20x2 (+5%) -> 19x3 (+6%)

20x3 (+10%) -> 19x5 (+12%)

20x4 (+15%) -> 18x4 (+18%)

19x2 (+10%) -> 18x3 (+12%)

18x2 (+15%) -> 16x2 (+15%)

That follows the general rule that "crits are larger and less common".

Now, let's try the "width boosted by 1" version of keen:

K20x2 (+10%) -> 18x3 (+12%)

K20x3 (+20%) -> 18x5 (+24%)

K20x4 (+30%) -> 17x4 (+30%)

K19x2 (+20%) -> 17x3 (+20%)

K18x2 (+30%) -> 15x2 (+21%)

Hmm. Works well, except for the keen scimitar. What if we special case that?

K20x2 (+10%) -> 18x3 (+12%)

K20x3 (+20%) -> 18x5 (+24%)

K20x4 (+30%) -> 17x4 (+30%)

K19x2 (+20%) -> 17x3 (+20%)

K18x2 (+30%) -> 16x3 (+30%)

There, it works reasonably well!

So, here is a model:

Ie, keen adds 1 to the width of your crits, unless your weapon is an x2 weapon. In that case, it instead boosts your crit damage multiplier by 1.Code:Old Standard Keen 20x2 19x3 18x3 20x3 19x5 18x5 20x4 18x4 17x4 19x2 18x3 17x3 18x2 16x2 16x3

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Note that 'rolling to confirm on a 1d20' is both ugly and not required. Just roll 2d10 to confirm as well, it is cleaner.