Die Roll Probabilities in GamingDate: 01/06/2001 at 10:52:43 From: Wray Ferrell Subject: Odds of winning battles I am designing a game based on the Rise of Rome. Battles are resolved by having both sides roll 2D6 with modifiers from better leaders, troops, terrain, etc. To simulate the ancient battles where one side should have been routed but was not, the attacker always loses if they roll a 2 or if the defender rolls a 12. Ties go to the defender. After modifiers the attacker must roll higher than the defender. My playtesters wanted a chart giving the odds of victory based on modifiers. I know the odds of rolling a 2 are 1/36 and the odds of rolling a 12 are 1/36. So the attacker has 1/18 chance of losing regardless of the die roll modifiers. It also seems that without any die roll modifiers the odds of the attacker rolling higher than the defender are 1/2. However, my math falls down when trying to determine the additional probability gained or lost from the die roll modifiers. I think the equation is: Prob. of Victory = 50 - 1/18 + (A * modifier) Thus a negative modifier, a desperate attack, subtracts from your chances, while a positive modifier helps them. My question is; how do I determine A? Thanks, Wray Date: 01/08/2001 at 14:02:28 From: Doctor TWE Subject: Re: Odds of winning battles Hi Wray - thanks for writing to Dr. Math. Your odds of the attacker winning aren't quite right for two reasons. First, you've counted the probability of the attacker rolling a 2 AND the defender rolling a 12 twice. (Once as the attacker rolls a 2, and again as the defender rolls a 12.) Second, you haven't accounted for the fact that ties go to the defender. The correct probability should be 575/1296 ~= 44.37%. As to modifiers, the amount they add to (or subtract from) the probability is not linear - it depends on how many modifiers you already have, and how many modifiers your opponent has. It also depends on how they are implemented with respect to the 2 and 12 rules: does a modified 12 (after additions) guarantee the defender wins, or only a "natural" 12? Same for the attacker. Finally, are all modifiers bonuses (i.e. they can only add to the attacker's or defender's rolls), or are there negative modifiers as well? Here are the probabilities of rolling each combination on 2D6 (combinations are out of 36): Roll Comb. Prob. ---- ----- ----- 2 1 2.8% 3 2 5.6 4 3 8.3 5 4 11.1 6 5 13.9 7 6 16.7 8 5 13.9 9 4 11.1 10 3 8.3 11 2 5.6 12 1 2.8 So here's the attack matrix: Attacker Roll: | 2 | 3 4 5 6 7 8 9 10 11 12 | Total Comb: | 1 | 2 3 4 5 6 5 4 3 2 1 | Att Def -------+----+----------------------------------------+--------- R C | | | 2 1 | 1 |\ 2 3 4 5 6 5 4 3 2 1 | 35 1 | | +-+ 3 2 | 2 | 4 \ 6 8 10 12 10 8 6 4 2 | 66 6 | | +-+ 4 3 | 3 | 6 9 \12 15 18 15 12 9 6 3 | 90 18 | | +-+ D 5 4 | 4 | 8 12 16 \20 24 20 16 12 8 4 | 104 40 e | | +-+ f 6 5 | 5 | 10 15 20 25 \30 25 20 15 10 5 | 105 75 e | | +-+ n 7 6 | 6 | 12 18 24 30 36 \30 24 18 12 6 | 90 126 d | | +-+ e 8 5 | 5 | 10 15 20 25 30 25 \20 15 10 5 | 50 130 r | | +-+ 9 4 | 4 | 8 12 16 20 24 20 16 \12 8 4 | 24 120 | | +-+ 10 3 | 3 | 6 9 12 15 18 15 12 9 \ 6 3 | 9 99 | | +-+ 11 2 | 2 | 4 6 8 10 12 10 8 6 4 \ 2 | 2 70 | +----------------------------------------+--------- 12 1 | 1 2 3 4 5 6 5 4 3 2 1 | 0 36 -------+---------------------------------------------+--------- Att. | 0 2 9 24 50 90 105 104 90 66 35 | 575 Def. | 36 70 99 120 130 126 75 40 18 6 1 | 721 Based on the assumption that (a) only a "natural" 2 or 12 guarantees the defender wins, and (b) there are no negative modifiers, here is a partial chart of the probabilities based on modifiers. You can extend it yourself if these assumptions are incorrect. Net | Attacker | Defender Mod | Comb Prob. | Comb Prob. ----+------------------------------- +9 | 1225 94.52% | 71 5.48% +8 | 1221 94.21 | 75 5.78 +7 | 1209 93.29 | 87 6.71 +6 | 1184 91.36 | 112 8.64 +5 | 1140 87.96 | 156 12.04 +4 | 1070 82.56 | 226 17.44 +3 | 974 75.15 | 322 24.85 +2 | 855 65.97 | 441 34.03 +1 | 719 55.48 | 577 44.52 +0 | 575 44.37 | 721 55.63 -1 | 435 33.56 | 861 66.44 -2 | 310 23.92 | 986 76.08 -3 | 206 15.90 | 1090 84.10 -4 | 126 9.72 | 1170 90.28 -5 | 70 5.40 | 1226 94.60 -6 | 35 2.70 | 1261 97.30 -7 | 15 1.16 | 1281 98.84 -8 | 5 0.39 | 1291 99.61 -9 | 1 0.08 | 1295 99.92 -10 | 0 0 | 1296 100 I hope this helps. If you have any more questions, write back. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/ Date: 01/09/2001 at 16:29:09 From: Wray Ferrell Subject: Re: Odds of winning battles First of all, thanks for the response. I do have a few more questions if you don't mind. I'm trying to understand the math rather than just get the answer. >Based on the assumption that (a) only a "natural" 2 or 12 guarantees >the defender wins ...Correct. >(b) there are no negative modifiers ...There is a -2 modifier for attacking over mountains or rivers, but I am not sure how that would affect the numbers. What is the difference between the attacker having +5 for having five combat units (+5 for 5 combat units) or having +5 for having seven combat units attacking over the mountains (+7 for 7 combat units and -2 for attacking over mountains)? Also, I understand from the chart you provided how you came up with the 44.37% chance of victory for an even battle. But how did you calculate the effects of the modifiers? I mean did you just "brute force" it by looking at the matrix? Thanks, Wray Date: 01/09/2001 at 17:48:19 From: Doctor TWE Subject: Re: Odds of winning battles Hi - thanks for writing back. >...There is a -2 modifier for attacking over mountains or rivers, >but I am not sure how that would affect the numbers. What is the >difference between the attacker having +5 for having five combat >units (+5 for 5 combat units) or having +5 for having seven combat >units attacking over the mountains (+7 for 7 combat units and -2 for >attacking over mountains)? Actually, there's no difference. I hadn't evaluated the situation that thoroughly at the time I wrote that assumption. Once I derived the chart, I realized that negative modifiers to the attacker have the same effect as positive modifiers to the defender, and vice-versa. The probabilities never go beyond those listed for +9 or -10, however. (Beyond this either the defender automatically wins, or the defender only wins on a natural 2/12.) >Also, I understand from the chart you provided how you came up with >the 44.37% chance of victory for an even battle. But how did you >calculate the effects of the modifiers? I mean did you just "brute >force" it by looking at the matrix? Each +1 to the attacker (or -1 to the defender) shifts the diagonal line on the matrix down one place. The quickest way to calculate it is to add up the numbers that the line passed over, add this total to the attacker's combinations and subtract it from the defender's. Remember that the numbers in the leftmost column and bottom row always belong to the defender's combinations - they never switch to the attacker. Similarly, each +1 to the defender (or -1 to the attacker) shifts the diagonal line on the matrix up one place. Again, add up the numbers that the line passed over, but in this case, subtract this total from the attacker's combinations and add it to the defender's. Yes, I did this by "brute force." (I had a spreadsheet generate the matrix and the diagonal sums.) With a matrix of this size, it is far easier to do it this way than to generate complicated combinatoric formulas - these get very messy when comparing sums of random numbers. I hope this helps. If you have any more questions or comments, write back again. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/ |
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