*The*guide for the gifted and mathematically challenged alike

Warhammer is a game of dice. In general, mathhammer aims to predict the dice. Its fault is its failure to take any amount of "luck" into account.

Lets begin with the basics. Flip a coin. Did it come up heads? A coin that can only come up heads or tails has 50% chance of coming up heads. Were you lucky?

Ok now flip two coins. I bet you got one heads and one tails, or two heads, or two tails. Not much of a bet is it? The fact is, all of these outcomes are possible, and all these outcomes are the only possible outcomes.

If you were asked to predict the flip of two coins what would you say?

Each outcome has a probability. You could predict that the flip will be one heads and one tails, but this only has a 50% chance of happening.

When we roll dice, you can try to predict the outcome just that same. If you have 4 BS3 markerlights, the prediction will be 2 markerlight hits. But just as with the coin flip, the prediction might not come true. And in fact, in this case, it will probably be something other than 2 hits.

In general, when trying to predict the number of successes in a dice roll, you multiply the probability of success of a single die by the number of dice.

I am told that the more formulas I write in this article the less popular it will be, but I hope if I keep them simple then it won't be too bad.

Predicted # Success = P(success) * # dice

This is not the whole picture. This formula does not take luck into account.

Probability mathhammer is able to model luck

Any number of shots are distributed randomly. Here's how to get the probability distribution:

P(# success) = p^r * (1-p)^(n-r) * nCr

P(# success) => The probability of getting that exact number of successes

p = P(success) => The probability of success for a single die

n = # dice => The total number of dice you roll

r = # success => The exact number of successes

nCr is the binomial coefficient, you will find it on your calculator. Hit n (4 for example), then the nCr button, then r (2 for example).

Ok lets put this to the test with our 4 markerlight example.

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`# success - P(# success)`

0 - 0.0625

1 - 0.25

2 - 0.375

3 - 0.25

4 - 0.0625

Thus there is a 0.375 probability of 2 hits.

Ok back to basics for a moment. The chance of rolling a 5 is 1/6, and the chance of rolling a 6 is 1/6. You can add these probabilites to get the chance of rolling a 5 or a 6, 1/6 + 1/6 = 1/3.

You can also add the probabilities in the distribution. The chance of 2+ markerlight hits is: 0.375 + 0.25 + 0.0625 = 0.6875.

Vehicles

Vehicles can only die once. The mistake people make when calculating the probability of killing a vehicle is to double count the probability of killing it twice. This is how to get the probability to destroy a vehicle:

1 - {1 - [P(success)]}^# dice

**For example**: Two deathrains shoot at an open topped AV10 skimmer that's moved fast.

Correct way to do it:

1 - {1 - [P(success)]}^# shots = 1 - {1 - [8/9*2/3*1/2]}^4 = 1 - [1- 8/27]^4 = 0.755

The mistake: Probability of killing the skimmer is

**not**:

P(success) * # dice = (8/9*2/3*1/2) * 4 = 1.19 ~ this is

**wrong**.

Lets not forget that we also have the chance of stopping the vehicle from shooting. It will be useful to know how to work out the probability of doing nothing to a vehicle.

P(nothing) = [1 - P(success)]^# shots = [1 - 8/9*2/3]^4 = 0.0275

Dumbing it down

All that stuff can get quite confusing for the average person. It can get very frustrating for both the mathhammer and non mathhammer parties when discussing the effectiveness of units. It is often the mathhammer parties that are at fault. They use simplified models that are easy to understand, but are not very good. Experience counts for more than how many space marines you will statistically kill.

One concept that is particularly hard to grasp is turning shots at a unit into probabilities. Lets use an example. 5 Space Marines with bolters shoot at a unit of Chaos Space Marines. You pick up 5 dice, and roll to hit. You pick up the hits, then roll to wound. Finally for each wound the Chaos player rolls an armour save.

Unfortunately that was the easy part. Ok to get the actual chance of a single bolter killing a Chaos Space Marine, you multiply the probability of each event together.

P(success) = P(hit) * P(wound) * P(failed save)

If we try to predict the number of dead space marines using the very first formula in this article, we get 5/9. Cleary, we cannot kill 5/9 marines. Clearly 5/9 is misleading when the Chaos player fails 3 saves! This is the flaw of regular mathhammer. This is the beauty of probability mathhammer.

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`# success - P(# success)`

0 - 0.555

1 - 0.347

2 - 0.0867

3 - 0.0108

4 - 0.000677

5 - 0.0000169

Instead of rolling the dice three times, it is equally valid to roll one dice. In this case it would have to be a D9. Roll 5 D9's and all 9+'s will cause casualties. This has the same distribution as doing it normally.

Ok now imagine a D1000000, a one million sided dice. You could instead of rolling the 5 D9's, roll one D one million. If it is 999983+, you kill 5 marines. If it is between 999324 and 999983, you kill 4 marines. Similarly, divide the D one million into 6 total sections each representing the probability for each number of kills. If you roll less than 555000, you get zero kills.

This is exactly the same as rolling 5 dice, picking up hits, rolling the hits, picking up wounds, then for each wound the Chaos player rolls armour saves, and each failed save a Chaos Marine is removed as a casualty.

In this way, it is much easier to turn the shots into probabilities. You're just rolling one very large dice (physically impossibly large). Just like you have a 1/6 chance to roll a 6+, you have a 445000/1000000 chance to roll a 555000+ (and thus kill at least one marine).

Conclusion

I understand that no matter what I do, some people just won't get it. Some people will still not see the benefit of probability mathhammer.

What I am doing is

**not**regular mathhammer. It is

*different*. I often feel as though everyone is like "mathhammer does not take luck into account". This

**does**take luck into account.

Clearly if this was mathhammer that is a contradiction, and unless you're not good at maths or English you understand how silly that stance is.

Either mathhammer

*can*take luck into account (in which case, if it does, it's clearly better), or this is

*not*mathhammer. It cannot be both.

As they say, single, attractive, mentally stable. Chose two.