Hello 3++ community, T2-Keks here with some math :P
I have been enjoying 3++ and other blogs for some time and read a lot that helped me improve my play and get new, interesting views on stuff. When I was about to talk to my local gaming group about Mathhammer, I looked up the various Mathhammer articles here on 3++. I liked them but I thought they aimed more towards people already knowing how to do the Mathhammer. An introduction that explains how it’s done is missing. So I wrote one up for my group. Since I took quiet some insights from 3++ I now try to give something back by contributing. So I translated my article and sent it to Kirby (original version in German can be found on our group’s blog: bockige-wuerfel.de). Pictures are done by our artist Timo!
You use the knowledge that each pip on a dice comes up equally likely. Then you check how many pips lead to success or failure on a given test. The quotient of number of possible successes to the total number of possible outcomes is the probability to be successful in that test.
A model with BS4 hits its target on a 3,4,5,6. Thus 4/6 possibilities are considered successes. 4/6 = 0.67 aka 67%.
Now if you want to chain partial results together to get a certain event, you calculate the probability for each roll of any die to be successful and then multiply these probabilities with one another to get the overall probability of that event to happen.
For example the chance to kill of a S4 shot fired with BS4 at a target with T4 and a 3+ save is calculated as follows:
To hit: 4/6 pips are successful
To wound: 3/6 pips
Fail the armor save: 2/6 pips
4/6*3/6*2/6 = 0.11 aka 11% is the chance of this to happen.
Graphical this means that you draw a tree that branches at every step between success and failure and has probabilities assigned to every branch. To calculate the probability of an event you follow a branch and multiply all probabilities you come across.
The three ways to fail (don’t hit, don’t wound, target saves successfully) each have their own chance to occur. If you add them together you get the total chance to fail: 89%. In sum with the chance to succeed (11%) you end up with 100%. That is important. All possibilities combined must give you a 100% chance. If you end up with more or less you are certain to have made a mistake. But getting a sum of 100% does not guarantee that you didn’t make a mistake ;)
If you include events that only occur under certain circumstances into such chains (like the reroll if the to-hit-test for twin-linked weapons), you have to carefully watch out to make the correct branches.
The example from above, this time with a twin-linked gun, looks as follows:
The roll to hit is just repeated if it failed initially. Two ways lead to a successful elimination of the target. Their chances are added up to get the total probability of a successful kill.
When interpreting this result one could say “A space marine has an 11% chance to shoot another marine with his boltgun.” (tree1). That is correct but formulating it this way easily leads to mistakes. If you put it like this: “One S4 shot fired with BS4 at a target with T4 and a 3+ save will, on average, net 0.11 dead targets.”, you are on the safer side. This statement can be scaled up correctly: After 18 shots fired one can expect two dead marines (18 fired, 12 hit, 6 wound, 2 failed armor saves). But 18 shots fired won’t get you a 200%-chance to kill a marine. 18*11% = 200% is wrong. More on that later may follow in a later post.
In a nutshell:
- check for every dice how many pips are failures and successes on a given test
- formulate quotient of “number of possible successes”/”number of overall possibilities”
- multiply these quotients along a branch of the tree to get the chance of success
- if there are multiple ways to succeed calculate them all and add up their individual chances to get the overall probability to be successful
- these results are averages not guarantees