Re-roll 1's to hit or wound?

In some circumstances in the game you have a choice of re-rolling 1s to hit or 1s to wound. For example, an Astra Militarum Office can use the “Take Aim” or “Bring It Down” order. Which is better?

We can answer this with some algebra.

First, let

pH = probability of a hit

pW = probability of a wound

Since a 1 would normally always be a failure, rolling to hit with re-rolling 1 becomes

pH + (pH × ⅙) = pH × 1⅙

Similarly rolling to wound with re-rolling 1 becomes

pW × 1⅙

The overall probability of a wounding hit with no re-rolls is

pH × pW

The probability of a wounding hit with re-rolling 1s to hit is

pH × 1⅙ × pW

and the probability of a wounding hit with re-rolling 1s to wound is

pH × pW × 1⅙

Due to the commutative and associative properties of multiplication we can swap around the terms and see that

pH × 1⅙ × pW = pH × pW × 1⅙

So the result is that it doesn’t matter whether you re-roll 1s to hit or 1s to wound – the overall probability of success is the same.

Note that this only works when you’re re-rolling only 1s. Also, there may be other reasons why re-rolling hits works better – for example if you’re overcharging plasma weapons.

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