Does my firebase benefit more from rerolling 1s to hit, or rerolling 1s to wound? In practical terms, this is the captain or lieutenant discussion (and might just be about to become more Cult relevant if some of our auras change as the previews suggest).
This is one of those questions that seems to really mess with people’s thinking. Almost everyone I’ve brought this up with tells me that I’m wrong, or that I’ve not taken certain factors into account. One view is that a captain should be used to support something like big guns, where the wound roll will take care of itself; others suggest that the lieutenant is better in that situation, as insurance against the sting of failed 2+ to wound rolls.
But as it turns out, it doesn't matter. They both have the same effect.
THE MATHS
The maths behind this is relatively simple. Reroll 1s always produces a 16.67% increase in accuracy (from a maths/theoretical point of view) – you can see the workings in the table below. We can also represent this with the fraction 7/6, showing an improvement of 1/6 over the baseline [with 1/6 and 16.67% being the same thing].
We can see that the more accurate a model is, the more it benefits from the reroll 1s effect. But it’s a proportional thing: picking up an extra 5 hits to add to the 30 you already had (from your BS2+) is the same, proportionately, as adding 1 bonus hit to the 6 hits you managed with BS6+. So whilst the absolute gain differs, the percentage gain is constant.
HOW DOES THAT WORK?
When we come to apply this to the game, we’re looking at three factors – the chance to hit, the chance to wound, and the reroll modifier. The first two will vary between 1/6 to 5/6; the reroll modifier is fixed at 7/6.
For example: hit on 3s, wound on 4s plays out as 4/6 × 3/6 or 12/36. To add the effect of a reroll 1s aura, we multiply that result by 7/6, meaning that 12/36 improves to 14/36. But because we're just multiplying three fractions together, the order doesn't matter.
It's all to do with the fact that a series of multipliers can be arranged in any order to produce the same outcome. For example, 2x3x4=24; but so does 3x4x2, or 4x2x3. However you arrange that sequence of numbers, it will always come to 24.
So in the previous example, any sequence of 4/6 x 3/6 x 7/6 will produce 14/36. It doesn’t matter whether that 7/6 comes in after the 4/6, or after the 3/6, or even if we put it first. The sequencing is unimportant.
In other words, it makes no difference whether we modify the to hit or to wound roll. So a captain and a lieutenant have the same effect (outside of outlying cases like plasma, where the specific overheating rule interacts differently with reroll 1s).
EXAMPLES - MATHS
Let’s look at some practical examples. Imagine that you have a venerable dreadnought firing an assault cannon at a T8 target. 2+ to hit (5/6 chance), then 5+ to wound (2/6 chance). So the basic chance to cause a wound [ie force a saving throw] is 5/6 x 2/6, or 10/36.
Now let's apply the captain's reroll 1s aura. This adds 7/6 into the maths, just after the roll to hit: 5/6 x 7/6 x 2/6, or 70/216. What if we use a lieutenant instead? It would be 5/6 x 2/6 x 7/6, which is still 70/216.
Or let’s take a less extreme example, and have marines firing at marines. This gives us 3+ then 4+, which is 4/6 x 3/6, or 12/36. With a captain: 4/6 x 7/6 x 3/6, or 84/216. With a lieutenant: 4/6 x 3/6 x 7/6, or 84/216. Exactly the same.
Volume of fire doesn't change things much either - it's just another number to add to the sequence. 10 marines rapid-firing bolters gives 20 shots. Which gives us either 20 x 4/6 x 7/6 x 3/6 or 20 x 4/6 x 3/6 x 7/6. Same thing either way.
EXAMPLES - DICE
This time, let’s use imaginary dice rather than fractions. Let's say I have 36 of these dice. Whenever I roll them, they always produce an exactly even distribution - 6 6s, 6 5s, 6 4s and so on. And let's use them to roll out 36 bolter shots against other marines, supported by either a captain or lieutenant.
Captain first. I start with 36 dice. I roll, needing 3s, and 24 are hits, 12 are misses, of which 6 are 1s. I reroll the 1s (which produce one of each number), gaining a further four hits, for a total of 28 hits. I roll those 28 dice for wounds, needing 4s. 14 come up as wounds.
Now the lieutenant. I start with 36 dice again. I roll, needing 3s, and get 24 hits. I have no reroll to hit this time, so take those 24 dice forward to the wound roll. I roll 24 dice to wound, needing 4s. 12 come up as wounds, and 4 come up as 1s. I reroll those 4 dice, gaining two extra wounds, for a total of 14 wounds.
Either way, we end up with 14 wounds from 36 shots. The captain carries more hits forward, but the lieutenant does more work with the hits that he has. It balances out in the end.
CAVEATS
The only difference is when other rules come into play. Plasma guns are a great example, because the captain's reroll affects their chances of blowing up (reducing it from 1/6 to 1/36), whereas the lieutenant has no effect on this. With plasma, the captain is a much better option. [In a single round of shooting, the outcome in terms of wounds caused would still be the same. However, the captain would have kept more plasma gunners alive for future round, which is why he's better here.]
We see another difference with exploding 6s. If you gain any benefit from 6s to hit, then the captain is a better option, as the reroll means rolling more dice at the ‘to hit’ stage. Oddly, though, if you have an exploding 6s effect on your ‘to wound’ roll, both the captain and the lieutenant have the same effect (but in slightly different ways). In the dice example above, the captain scores 28 hits, so rolls 28 dice to wound; the lieutenant scores 24 hits, but will (on average) reroll another 4 dice, also rolling 28 dice to wound overall. This discrepancy occurs because the captain effectively goes into the lead at the ‘to hit’ stage, with the lieutenant catching up at the ‘to wound’ stage. So a ‘to hit’ effect benefits from the captain’s rerolls, but it’s all evened out by the time the ‘to wound’ stage comes up.
All of this math is also predicated on using either the captain or the lieutenant on the same unit. If you're choosing between different units to support with a single character (such as aggressors or intercessors in your example), then the decisive factor is now the differing output of the units. But that's a whole other kettle of numbers.