This is a topic I touched on briefly before, but I think it’s time for a more comprehensive review. As I’ve said previously, many players have bad instincts for how odds ‘stack up’, and I often hear comments like ‘if a terminator has to make 6 saves, you’d expect one to fail’. Time to challenge some assumptions.
A squad of terminators come under bolter fire and have to take 6 saves, what is the probability that they take no casualties? A squad of marines come under a similar hail of bolter fire and also have to take 6 saves, what is the probability that they take no casualties?
At tournaments, and during club play, you may often hear cries of consternation as someone can’t believe that their opponent just made X number of saves in a row. I think this ties in to certain types of (erroneous) expectations. Nothing in this game of dice is certain; and you can never get a 100% guarantee of success.
That sounds pretty trite, and taken purely at face value, it is. But I’m trying to get at something a little deeper. If a terminator has to make ten saves, or a thousand saves, there is always a chancing of making it, but the odds don’t scale in the way that people expect. Simply put: a thousand bolter shots is not one thousand times more likely to kill a terminator than one bolter shot; nor is ten bolter shots ten times more likely. As the number of saves to be taken increases, the odds of survival go down in an exponential way (for the non math types that means they start high, and gradually get lower but never quite reach zero).
Here’s one I prepared earlier:
This graph shows how terminator armour and power armour behave as the number of saves to be taken goes up. Terminator saves are in red, and marine saves are in blue. The x-axis represents the number of saves that have to be made, and the y-axis shows the odds of the squad taking no casualties for the corresponding number of saves.
So this gives us the answer to the opening question. If the termies have to take 6 saves, then there’s a 33% chance of them taking no casualties (i.e. find 6 on the x-axis and then look at the corresponding point on the y-axis for the red line). The other group aren’t so hot, if the marines have to take 6 saves, then there’s a 9% chance of them taking no casualties.
If we extend the analysis a bit, the marines have only a 1% chance of taking no casualties from 12 saves, whereas the terminators have a more respectable 11% chance. Even at 18 saves the terminators still have a 4% chance of walking away without a scratch. Now 4% may sound like very long odds, but in truth its not far off the odds of getting a ‘perils of the warp’ result for a psyker. So not something you’d see a lot, but hardly beyond belief.
One (slightly esoteric) point to note is that this is a ‘memoryless’ system. This gets a bit subtle, but what I mean is that the current odds aren’t affected by what happened before. So if a squad of terminators all survived 6 saves last turn and are now facing 6 more saves in this turn, the odds don’t stack to 11% (i.e. for 12 saves), they stay at 33% (for 6 saves). Whatever happened in the past doesn’t affect your current action.
Now that we’ve covered some specifics, I’ve taken the liberty to graph the behaviour of saves from 2+ to 6+ when having to make up to 6 saves in one block. Each coloured line represents a corresponding type of save from red for a ‘terminator’ save, blue for power armour, and so on through to the grey line for a 6+ save.
This follows the same pattern as the previous, it just shows more armour types. But as an interesting illustration, based on the graph check out the following:
A terminator squad taking 6 armour saves has a 33% probability of taking no casualties
A terminator squad taking 6 storm shield saves has a 9% probability of taking no casualties
A terminator squad taking 6 cover saves (4+) has a 2% probability of taking no casualties
A terminator squad taking 6 invulnerable saves (5+) has a 0.14% probability of taking no casualties
Do keep this in mind the next time you fire your hydras at my obliterators!