KOW - Fat Tails

On the Fanatics FB group recently there has been an upsurge in discussion about mathhammer (or expected outcomes when rolling dice). There seems to be a real lack of knowledge regarding the maths of probability and what the relevance of expected out comes are.

I thought I'd show a couple of things that you should be aware of, because in considerations of probabilities "A little knowledge is a dangerous thing". And just to give my thoughts some cred, I traded financial derivatives for twenty years and saw fortunes lost because people didn't understand such concepts as fat tails, long tails, volatility "smiles" and skews. The Global Financial Crisis was largely caused by people believing the numbers outputted from their "black box" computers without understanding the maths or assumptions behind both the inputs and outputs.

Just Remember "It's all about the Greeks"

When you roll two D6 dice you have a range of outcomes from 2 to 12. You have an expected outcome of 7, with one chance in 36 of getting a total of 2 and the same of getting a total of 12. The important thing here is that your expected outcome is 7 although that only has 6/36 chances of being the actual outcome. 

What you in fact have is an expected outcome "centred" at 7 that follows a normal probability distribution pattern.

This is the pattern we all know and love - and should be make out wargaming decisions based on. In most cases. The important thing here is that this is a normal distribution pattern with no skew - and very importantly clear defined boundaries 2 and 12.

However this is not always the situation that confronts us. Let me use an example utilising my combat calculator "The Abacus of War":

Your opponent's Ratkin Tunnel Slave Horde has 30 25 attacks. It hits on 5+ and has no modifiers to hit or wound. Against a unit with Defence 5 it should do 2.8 wounds. However that doesn't tell you the full story. If he roll badly he might do 0 wounds, right? But if he rolled well then conceivably you could be wounded up to 25 times.

It's not likely to happen but it is possible.

Looking at the maths there is approximately a 90% probability he'll do at least 1 wound, around 70-75% that he'll do at least 2 wounds and 50% he'll do 2.8 wounds. But what about the other side?

This is where we have a skew and a fat tail.

16% (or 1 in 6) he'll do over 4 wounds, one in 20 times he'll do 6 wounds and one in 100 he'll do almost 8. There is an infinitesimally small chance that he'll do 10+ wounds but a chance exists all the way up to 25 wounds.

The above is an example of what you would expect to see (ignore the axes). In our example the hump of the curve is centred on 2.8 while the left hand side of the curve intersects (as above) at zero. The important point is that the tail extends all the way to 25.

One thing in our favour is that the tail can be calculated - because it is a closed system of 2 dice roles with specific obtainable outcomes (0 to 25). In real life (finance, economics etc) things aren't that simple ..... that's why you seem to get 100 year floods every 10 years!

The overall takeaway is that be careful what you "expect". Mathhammer is a great tool but be aware of the underlying assumptions.