Dice pools, now with added spreadsheet
So I spent a little bit tossing dice pool data into a spreadsheet (available in Openoffice.org ODT and Excel XLS formats).
First of all, I found a trend to the number of successes. It's a normal distribution, as should be evident from the roller graphs. However, the distribution peaks at
pool ---- 2for static rolls and at
( attacker pool - defender pool ) --------------------------------- 2
for contested rolls. This is an easy way to calculate the weighted mean roll. However, in all circumstances, the distribution becomes wider with larger pools, giving a lower chance of any given roll.
weighted mean roll = sum( chance of rolling X successes * X ) for X = ( -2 * defender's pool or 0 ) -> twice the pool
As it's a normal distribution, the weighted mean is the number where 50% of your rolls are above it and 50% are below, and provides a relatively handy way to compare different tweaks to a roll.
For instance, the net result of rolling a contested roll against an opponent with four dice is equivalent to having a 4-die penalty on your roll, except that there is still a chance of success at levels below 5.
This then means that mean damages can be calculated, which I did for three kinds of weapon damage1 and three kinds of defence.2
So, that all said, let's re-join Gord and Sandeep, along with their pilot, Avril (pool of 6 in attack and defence). A few presumptions here: They are still facing a horde of 4-die mooks. Each of them is carrying an azoth set to deal 8 damage.
NB: The mean damage dealt by the characters looks a little low. That's because it already factors in the possibility that they won't hit. So it's not 39.1% chance of hitting for 1.93 damage. The two are independent of each other; the mean damage is over all combat rounds, not just the ones in which they hit.
If they are attempting to split their dice pool and both attack and defend, here is how they fare: Attack Mook 1 Mook 2 Mook 3 Mook 4 Mook 5 Character Gord (4) Chance of hitting: 39.1
Mean damage: 1.93 Chance: 87.0
Total: 87.0 Chance: 87.0
Total: 98.3 Chance: 87.0
Total: 99.8 Chance: 87.0
Total: 100.0 Chance: 87.0
Total: 100.0 Gord cannot do this, so he instead attacks. This means that the mooks have their usual chance of hitting him. Avril (6) Chance of hitting: 28.7
Mean damage: 1.38 Chance: 74.2
Total: 74.2 Chance: 87.0
Total: 96.7 Chance: 87.0
Total: 99.6 Chance: 87.0
Total: 99.9 Chance: 87.0
Total: 100.0 Avril has enough of a pool to attack once and still be able to defend with one die. She does that on the first attack, then gets the base chance of being hit on the other 4. Sandeep (9) Chance of hitting: 58.7
Mean damage: 3.06 Chance: 39.2
Total: 39.2 Chance: 61.1
Total: 76.3 Chance: 87.0
Total: 96.9 Chance: 87.0
Total: 99.6 Chance: 87.0
Total: 99.9 Sandeep has enough of a pool to attack once and defend twice before he is out of dice. However, if faced with more than two thugs, he is relatively assured of being hit.
Now, what happens if the three of them go on full defence? Attack Mook 1 Mook 2 Mook 3 Mook 4 Mook 5 Character Gord (4) Chance of hitting: 0.0
Mean damage: 0.0 Chance: 49.8
Total: 49.8 Chance: 61.1
Total: 80.5 Chance: 74.2
Total: 95.0 Chance: 87.0
Total: 99.3 Chance: 87.0
Total: 99.9 In full defence, Gord dramatically improves his chances of dodging the first two mooks, but by the third, only has a 1/20 chance of not being hit. Avril (6) Chance of hitting: 0.0
Mean damage: 0.0 Chance: 30.7
Total: 30.7 Chance: 39.2
Total: 57.9 Chance: 49.8
Total: 78.8 Chance: 61.1
Total: 91.8 Chance: 74.2
Total: 97.9 Avril fares a bit better in full defence. She has reasonable, though slim, odds of dodging three, and it's not until five attack her at once that her chances drop to below 5%. Sandeep (9) Chance of hitting: 0.0
Mean damage: 0.0 Chance: 13.8
Total: 13.8 Chance: 18.3
Total: 29.6 Chance: 23.7
Total: 46.3 Chance: 30.7
Total: 62.8 Chance: 39.2
Total: 77.4 Sandeep, unsurprisingly, gets a severe boost to his dodging, with less than a 50% chance of being hit by three and almost a 25% chance of dodging five.
What if all the characters in this combat had static defences (equal to half their pool)? Attack Mook 1 Mook 2 Mook 3 Mook 4 Mook 5 Character Gord (4) Chance of hitting: 63.7
Mean damage: 3.04 Chance: 63.7
Total: 63.7 Chance: 78.0
Total: 92.0 Chance: 87.0
Total: 99.0 Chance: 87.0
Total: 99.0 Chance: 87.0
Total: 100.0 So far, this is the best scenario. Gord is able to attack, and can defend somewhat against one mook. Any more and he's almost certain to get hit, but this is the first scenario in which he can be at all effective. Avril (6) Chance of hitting: 87.0
Mean damage: 4.48 Chance: 40.2
Total: 40.2 Chance: 63.7
Total: 78.3 Chance: 78.0
Total: 95.2 Chance: 87.0
Total: 99.4 Chance: 87.0
Total: 99.9 Avril fares even better. She's almost certain to hit, and unlikely to be hit by the first mook. The second has decent odds of hitting her, though, and the rest are almost certain to do so. Sandeep (9) Chance of hitting: 97.3
Mean damage: 5.64 Chance: 0.0
Total: 0.0 Chance: 40.2
Total: 40.2 Chance: 63.7
Total: 78.3 Chance: 78.0
Total: 95.2 Chance: 87.0
Total: 99.4 Finally, Sandeep benefits most of all from this. He is very unlikely to miss, and two mooks are likely to miss him. Past two, however, he faces the same problems as the others.
In the spreadsheet, I outline a further set of six options, with varying levels of risk and differing curves between a 1 die pool and a 10 die pool.
However, the definite lesson here is that, without a system that favours the defender, chances of being hit quickly add up, and with five people attacking them, virtually everyone is likely to be hit.
Thus, I think that the '5 mooks' column is a bit of a red herring. Look at 2 and 3 mooks for a better portrayal of how well a given defence works.
Anyhow, this is just a sampling of the data. If you're browsing the spreadsheet for more clues as to how to ungame the system, the Results and Multiple Attackers sheets are likely the most valuable, as they're synopses that give an overall picture. The other sheets are one of two things: source data or charts of the overall data.
Enjoy.
- A 4 die weapon where you roll the damage plus your successes on the to-hit roll.
- A 2-damage weapon where you deal damage equal to 2 + your successes on the to-hit roll.
- A 4-damage weapon that does a fixed 4 damage no matter your to-hit roll.
- Defence roll of four dice.
- Static defence of 2, subtracting from the to-hit roll.
- Defence roll of four dice, roll subtracting from the attacker's pool.