Courting the Improbable

Warhammer 40K is, if nothing else, a game of dice. Random numbers generated in certain limited ways govern most of the actions through which we meaningfully interact with our counterparts. So its no surprise that managing probability has a central place in generalship.

Much of the chatter regarding the randomness inherent in our game is focussed on ameliorating or avoiding those times that probability falls out of our favour. For example, take the number of forum posters bemoaning the switch to random charge distances in 6th ed, and the common advice to work with it by getting as close as possible.

But, extremes of probability occur on either side of the mean. Sometimes the dice will simply fuck us. But other times, wild and unexpected success can happen - provided that we allow it to.

Take the random charge length. By getting as close as possible, we reduce the chance of an assault failing. There's nothing wrong with that, of course. But if the random charge length is only looked at through the prism of higher chances to fail, we rob ourselves of the upside; a much larger charge range.

Some units simply don't care about the negative consequences of a failed charge, generally exposure to overwatch fire for no gain, or a turn out of position. Some enemy units pose only a small threat to the charger, or none at all, while units like Dreadnoughts and Monstrous Creatures can often wade through an average unit's overwatch quite comfortably. These units have gained much from random charge length, as they can now legitimately threaten units up to 18" away at the start of their turn! And if you try these long-shot assaults where appropriate, sometimes they will come off - maybe with game-defining implications.

Unit-specific improbability
There are other ways to explore the same concept. One idea that's been kicking around in my head for a while is making sure the extremes of probability are in your favour.

How? Consider the Ork. He has next to no armour, and only very limited ability to hit with any shooting. Thus, we expect him to die when wounded, and to miss when he shoots. If he should actually make an armour save, that is a pleasant surprise! Likewise, any shooting hits. This means that there is much more room to err on the side of what we would call good luck rather than the limited scope for the converse.

For example, let's say a mob of twelve boys comes under heavy lasgun fire and takes twelve wounds. Our average result is two successful saves from those twelve wounds. A poor result would be only one or no saves - in this case the squad may be wiped out, but since it would have been reduced to two boyz remaining anyway, this is not such a big deal. On the other hand, if we get lucky we may end up with four or five boyz remaining - and that may be enough to achieve something, or at least cause another enemy to target the 'lucky' mob instead of moving on to a fresh threat. In this way, the extreme event helps our cause, while the other extreme is actually not so far away from the average.

Compare that to the Space Marine. We expect him to pass his armour saves, and hit what he shoots, and he is accordingly worth many more points than the Ork. However, this means that his extremes of probability are all on the 'unfortunate' end of the spectrum. Consider that the squad may take three wounds. We would expect one space marine to die. If we get a little lucky, none will die, which is nice. But if we get a little unlucky, two or even three may die, a much worse result than only one. Thus, whenever probability is noticeable for the Space Marine player, it is a negative. More men are dying than should. More shots are missing their target than should. This happens because there is more room on that side of the probability spectrum; and is perceived by the player as bad luck simply because the improbable scenario is the adverse scenario.


Do they average out?
Clearly, a 6 point Ork is an inferior unit to a 14 point (thanks, Dark Angels!) Space Marine, and the relative point costs reflect their difference in abilities. If we level the points, say by taking 7 Orks and 3 Space Marines (both 42 points) we can more appropriately compare them. I've assumed here shoota-equipped Orks and Tactical Space Marines, for convenience.

The most likely outcome, were they to exchange fire simultaneously, would be that no Space Marines die (44.9%) and 2 Orks die (32.9%). So the average outcome, unsurprisingly, favours the Space Marines.

It is often the case that larger samples produce results that are closer to average. In this instance, were either side to take three uninterrupted volleys at each other (say if the other unit was preoccupied with another battlefield role), their odds of annihilating their foe are pretty similar (41.5% for the Orks, 39.1% for the Space Marines). If there was room for only two volleys, the Orks have a 20.2% chance of wiping out their enemy; the Space Marines can claim only a 6.6% chance of doing the same. In one volley, although the Space marines hold the average advantage, they have no chance of destroying all of the Orks with only 6 shots, while the Orks have an improbable, but possible (3.9%) chance of extirpating all 3 of the Space Marines from the scenario.

In the 24 games that the Space Marines kill a pair of orks, not much of consequence happens. But the 1 game that the rabble of leftover Orks unexpectedly kills the rest of that damaged combat squad offers the potential of a significant reversal. If those three Space Marines happened to be holding an objective and are now dead, it could change the outcome of the game - in  a way that Space Marines thinning the ranks of leftover Orks rarely would.

Now, how many instances are there in a single game that this scenario could apply? If you've got 25 examples of units interacting with each other by shooting, engaging in melee combat - say 4 per game turn over six turns, with 1 tacked on somewhere - then a 4% chance (1/25) doesn't seem so unlikely to happen at least once in every game or two.

So where do you want your improbable events? Will you expose yourself to the chance that things will go very right? Or simply hope that they don't go very wrong...

(I wanted to have a look at some systemic improbabilities, like Overwatch, Seize the Initiative, etc - but this is already a bit of an essay. So I'll leave that for next time!)