Hammer of Math: Optimal Re-Roll Strategy for Critical-Hit Abilities

This week’s Hammer of Math takes a dive into the optimal time to use a re-roll when exploring the use of abilities like Lethal Hits in Warhammer 40k.

Critical Hit mechanics have become a major part of 10th Edition, to the point where abilities like [DEVASTATING WOUNDS] have required some tweaks to keep then in line. And while re-rolls were theoretically not going to be incredibly prevalent, it’s difficult to ignore how many Oath of Moment and [TWIN LINKED] opportunities are out there and giving players the option to pick up a die and toss it again.

With that in mind, one question that comes up is: Should you use the re-rolls to “fish” for critical results? Or put another way, when is the optimal time to re-roll a D6 when the test has been passed but the result wasn’t a critical success? For example, take a unit of Aggressors lead by an Apothecary Biologis who is giving the unit [LETHAL HITS]. Say they’re firing at a high-toughness target under Oath of Moment and can re-roll the hit roll. Given the unit’s Ballistic Skill and the target value for the wound roll, should they re-roll successful hits for a chance to score lethal hits and bypass the wound roll?

Deciding When To Re-Roll Your Successes

The math behind this question is pretty simple. The question is whether or not the cumulative probability of making both rolls is greater than the probability of succeeding on the second roll given a non-critical success on the first. In other words, is my chance of auto-wounding greater than my chance of hitting and wounding given that there’s a chance [LETHAL HITS] could activate and automatically wound? In the case of a BS 3+ Aggressor shooting a S4 weapon against a T12 Land Raider the answer is “No.” The chance of wounding the Land Raider is 17%. The chance of hitting and wounding the Land Raider is the chance of a Critical Hit (17%) plus the chance of hitting the target with a non-critical hit and wounding it (8%), or 25%. Since 25% is greater than 17% you would want to re-roll and try again. But what if there are re-rolls involved, or someone brings Bolter Discipline and now you have Critical Hits on a 5+? Don’t worry reader, I got your back with charts.

How To Use These Charts

Roll 1 is the roll you just made, while Roll 2 is the next roll up. For example, if you’re looking at the viability of re-rolling for [LETHAL HITS] then Roll 1 is the Hit Roll and Roll 2 is the Wound Roll. When looking at [DEVASTATING WOUNDS] then Roll 1 is the Wound Roll, and Roll 2 is the probability that the target will fail their Save (a Save of 2+ is equal to a 6+ to fail).

1. Figure out if the target value for a critical success is 5+ or 6 and use the appropriate chart.
2. Pick the sub-chart based on whether or not the second roll has no re-rolls, re-rolls values of 1, or re-rolls failures.
3. Select the row corresponding to the target value for Roll 1.
4. Select the column corresponding to the target value for Roll 2.
5. If the Re-Roll column is highlighted orange, then the chance of success is higher if you re-roll the success.

Example

Timmy, Primaris Lieutenant of the Astartes Ramrod Chapter, is leading a unit of Hellblasters against an Armiger which has an Oath of Moment dropped on it. The Hellblasters overcharge, as they are not cowards, and they have a BS of 3+ and a wound roll of 5+. Given that Timmy is giving his unit [LETHAL HITS], should they re-roll?

Achieving a Critical Hit is a 6+ so we use the first chart. There are no re-rolls on the wound roll so we use the first sub-chart. Looking at the 3+ row for Roll 1 and the 5+ column we see that the probability of success without a re-roll is 33%. With a re-roll it’s also 33%. Turns out it doesn’t matter. Do what feels good.

The Joys of Sustained Hits

Of course Warhammer 40k is nothing without random acts of synergy, and a perfect example can be found in combining [LETHAL HITS] with [SUSTAINED HITS 1] with a target value of 5+ courtesy of Bolter Discipline. So when dealing with those situations, what is the optimal strategy?

It turns out that the optimal strategy for Bolter Discipline is to just re-roll everything with a BS of 3+ or better, because the chances of increasing your output are significant. It’s also worth noting exactly how effective this combination is. For example, a normal BS 3+ shot with a wound roll of 6+ has an 11% chance of success. Add Bolter Discipline to provide Critical Hits (i.e. both [LETHAL HITS] and [SUSTAINED HITS 1] accounting for the Apothecary Biologis as well) on a 5+ and the chance becomes 44%. Add in a full complement of re-rolls and for every attack you make there’s a 69% chance the target will be making a saving throw. Nice.

The Final Word

There’s a lot to unwrap here so let’s go over some key points:

• When to apply a re-roll for a critically activated ability heavily depends on the target values for both the first and second roll, the target number for the critical hit, and the presence of re-rolls for the second roll.
• Logically, using a re-roll is the best choice when the first roll has a high probability and the second roll has a low probability. The more the probability of the second roll goes up (such as with re-rolls), the less likely it is that re-rolling the first roll is the optimal strategy.
• If you’re combining [LETHAL HITS] and [SUSTAINED HITS 1] you’re going to want to re-roll everything.
• Bolter Discipline is some powerful stuff.

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