Reading the recent post by Blazinghand on long-range anti-tank, I was reminded of one terminology issue that often creeps in when people mathhammer absolute and relative efficiency of weapons and units. And since I long wanted to write about it anyway, this may be a good time to finally do it

What I'm talking about is the use of the term "Expected" like this (highlights are mine):

Damage output:

1 Bright Lance shot, 1 PL giving 2 shots, 6 AEML/Reaper Launcher shots, all at pretty good BS, and with the Reaper-fired weapons suffering no penalties against Hard To Hit:

Total **Expected** Damage against T7, 3+: 9.4 damage

Total **Expected** Damage against T8, 3+: 7.1 damage

This is what I see as a significant mistake in the language: these are

**average** values rather than "expected" ones. Now, I know many people who tend to word it like this, and even those who don't specifically use the word "expected" still often tend to see the average as something that can be "expected" in each instance. This might seem a small thing, but imho it is important because it affects the way how people understand and interpret the numbers, and often leads to false expectations, which in turn sometimes result in people getting upset and frustrated by actual dice rolls.

Now, this is a mistake because, maths-wise, average of a random value is

**NOT** what can be expected in each specific realisation of that value. Average is just that - average, a value which you will hit (approximately) if you average out a very long series of instances of a certain random value. However, hitting the average in each particular instance is quite unlikely.

As you probably know, in addition to average a random value would also have standard deviation. The only thing that can be said with any degree of certainty about what can be "expected" of each specific instance of a random value is this: it will likely (as in "more often than not", or with ~68% chance) hit somewhere between (average - standard deviation) and (average + standard deviation). Anything within that range is pretty much equally "expected".

For example, 9 Dark Reapers firing Starshot at a standard vehicle will hit and wound on 3+, followed by 5+ save. This gives us the effective chance of ~0.296 to inflict an unsaved wound with each shot, and the average of ~2.67 unsaved wounds from 9 shots, which will multiply into ~8 points of damage.

But 9 shots with 0.296 chance of a wound per shot will also give us standard deviation of sqrt(9*0.296*(1-0.296)) ~= 1.37. This means that the actual number of wounds from such volley can be expected to be anywhere between 2.67-1.37=1.3 and 2.67+1.37=4.04. However, actual number of wounds cannot be fractional, so rounding to the nearest whole we get the result of anything between 1 and 4 wounds, dealing between 3 and 12 points of damage.

Have to admit, when I did such simple calculation for the first time, it was a bit of a shocking revelation to me - because these simple numbers give a clear explanation of why the 40k in general often feels so unpredictable and luck-dependent. Thing is, statistically the range of 1 to 4 unsaved wounds from 9 Starshots corresponds to a

**very small** deviation from the average. In other words, from statistics standpoint, getting 4 wounds means that your dice went

*just slightly above average*, while getting 1 wound means that your dice went

*just slightly below average*. But think of what a tremendous difference that would make in game terms! 12 damage means a vehicle destroyed outright, while 3 damage means a barely scratched paintwork - and both results are well within of what can be easily expected from each volley. If you never though of this before, I suggest you to roll this in your mind for a while...

This shows one fundamental design problem of the 40k as a game system: random deviations in the results of actual dice rolls which statistically qualify as really very small often have a disproportionally huge impact on the result of a specific action and on the whole tactical situation in the game. And that is precisely what often drives that subjective feeling of being "lucky" or "unlucky" - while in fact both may be well within the boundaries of "expected".

So, while we can (and should) calculate and use averages for our reference, they have little to do with what can be "expected" - because in actual situations we can expect... well, pretty much anything really

Thanks for reading!

P.S.

While what I wrote above is mostly academic, it leads us to at least one very practical conclusion:

When you need to accomplish something in-game (e.g. destroy a specific target), and the calculation shows that

*on average* you should succeed, this means that your actual chance of success is only about 50%. If you want something done more or less reliably, you should aim for an average that is significantly higher than the result that you need to succeed.