I found it fun to do this, and thought others might find it useful in planning strats; so here it is.
The formula for the average of an open ended roll
M= n(n+1) / 2(n-1)
The formula for the variance of an open ended roll
V= n(n+1)(n^2+7n-2) / 12(n-1)^2
n= the number of sides in both these formulas, in this case 6.
also, multiple dice are additive in the formulas, so doing them will return the results of a single d6, simply double them for a 2d6.
they solve out to:
M= 8.4
V= 3.2
now, the DRN will actually cancel out in all rolls, so all that matters is the difference in the constants. Thus, in an attack roll where the attacker has an attack 13 and the defender a defense of 14, there is a difference of 1 between the attacker and defender. divide this difference by the variance to get the z score. the z score can then be compared to a z table to see what the probability that the defender will block the attack is. below though i've listed all results for a difference up to 6. the percent chance is basically the percent that the unit with the lower number can roll a high enough DRN to win the roll. so a unit that adds up to being 6 less than the opponent on whatever number (attack - defense, fear - moral, missile - missile dodge, ect), will only have a 3% chance to win a roll.
X-Xbar:
6......3%
5......5%
4......11%
3......17%
2......26%
1......38%
0......50%
Lastly, note that this table assumes normality, which is not the case. The distribution oscillates in a pattern, but which is difficult to represent in a formula. The oscillation is due to the open ended roll, thus there is a higher chance to roll both a 6 and an 8 than there is a 7. There is effectively a 'hump' in the distribution at 6 then, meaning that the estimates above for a difference of 6 are slightly underestimated. Additionally, the distribution isn't normal, but log normal, and asymptotically approaches infinity. A log normal distribution requires recomputing everything with a log link, which I may do at some point in time to give more exact estimates. Basically, all estimates are slightly underestimated due to the long positive tail that the distribution should have. This long positive tail has a special effect in some checks where a 'win' for some actions is particularly meaningful. Such as, an attack that rolls unusually high most likely gets a guaranteed hit, as it is unlikely a defender will also roll an unusually high roll on the same check. A successful attack results in additional things that a successful defend does not. This makes the quality of log normal distributions to allow higher rolls particularly meaningful for most rolls in a way that can't be captured in even a log normal distribution. Rather, a Binomial or perhaps Poisson distribution would have to be used to model the effect of a number of success over a time period.
Other than all that though, it is a decent table. I post it again so it doesn't get lost in all that text
X-Xbar:
6......3%
5......5%
4......11%
3......17%
2......26%
1......38%
0......50%
probabilities of winning a DRN
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