When it's worth fishing for critical hits

[Paladin777]

Kind of what the title says, though I'm not really asking about it. 
 

Instead I've calculated when it's worth it to fish for critical hits when you've got rerolls and when not to and thought I'd share it with the class! This is specifically impactful for marines because of OoM.

 

For this exercise I want to show when it's worth fishing for critical hits, and when it's not, in regard to Lethal Hits, Sustained Hits.

 

To start I wanted to establish that I am referring to 'fishing' as the practice of rerolling all dice rolls that do not score critical hits, including normal hits, instead of just failures to hit. Of course this only applies where rerolls are allowed.

 

Sustained hits

 

Sustained hits are only mathematically worth fishing for when the sustained number is equal to or greater than your to-hit number. For example, sustained hits 2 is worth fishing if you have a 2+ to hit.

 

If your sustained hits number is 1 less than your to-hit number there's statistically no point to rerolling as your chance of hitting, including the sustained hits, is exactly 1. For example, inceptors have SH 2, which is one less than their 3+ to hit, which means that if you roll enough dice the number of total hits will equal the number of dice rolled. In other words, fish if you feel lucky, don't if you don't. The probability is that the results will be the same. 

 

If your sustained hits number is 2 lower than your to hit roll, it's not worth it. For example, a Heavy Bolter (with SH1) firing at BS 3+ is not worth fishing for.

 

Lethal hits

 

Lethal hits are a bit trickier as you need to factor in both the to-hit roll and the to-wound roll. If the chance of landing a successful wound from an attack with lethal hits (without rerolls) is greater than the chance of landing a successful wound per regular hit, then fishing is worth it. This is less often than you'd think. 

 

For each combination of to-hits and to-wounds, see the tables below. 

Spoiler

Lethals with a 2+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.7222	.8333	NO
3+	.6111	.66	NO
4+	.5	.5	Do you feel lucky?
5+	.3888	.333	YES
6+	.2777	.1666	YES

Lethals with a 3+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.5833	.833	NO
3+	.5	.66	NO
4+	.416	.5	NO
5+	.333	.333	Do you feel lucky?
6+	.25	.166	YES

 

Lethals with a 4+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.444	.8333	NO
3+	.388	.66	NO
4+	.33	.5	NO
5+	.277	.33	NO
6+	.222	.166	YES

 

Lethals with a 5+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.305	.833	NO
3+	.277	.66	NO
4+	.25	.5	NO
5+	.22	.33	NO
6+	.194	.166	YES

As you can see, for most marine units it's only worth fishing for Lethal hits when you're wounding on a 6+. 

 

 

Combined lethal and sustained hits. These are trickier still... and I really don't feel like going into detail about how I calculated the results!

 

Spoiler

With a 2+ to hit it's always worth fishing, but here's the numbers anyway

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.861	.833	YES
3+	.722	.66	YES
4+	.583	.5	YES
5+	.444	.33	YES
6+	.305	.166	YES

 

 

Lethal sustained hits with a 3+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.722	.833	NO
3+	.611	.66	NO
4+	.5	.5	Do you feel lucky?
5+	.388	.33	YES
6+	.277	.166	YES

Lethal sustained hits with a 4+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.583	.833	NO
3+	.5	.66	NO
4+	.41666	.5	NO
5+	.333	.33	Do you feel lucky?
6+	.25	.166	YES

 

Lethal sustained hits with a 5+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.44	.833	NO
3+	.388	.66	NO
4+	.33	.5	NO
5+	.277	.33	NO
6+	.222	.166	YES

 

So as you can see, even with both lethal hits and sustained going off, the average marine is still only going to benefit from fishing if they need a 5+ to wound. 

 

 

You may ask about the effect of twin-linking... im not going to do the math for these, but I think that would swing the numbers slightly in favor of not fishing. Anyone who wants to prove or disprove that can knock themselves out!

 

Moving on, let's see what happens when you throw 5+ crits in the mix. I'm not going to calculate sustained or LH individually as I don't think that's going to happen often enough to be worth my time. We want to know about lieutenants/biologis' with fire discipline!

 

Spoiler

5+ lethal and sustained hits with a 2+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	1.16	.833	YES
3+	1	.66	YES
4+	.833	.5	YES
5+	.66	.33	YES
6+	.5	.166	HELLS YES

 

5+ lethal and sustained hits with a 3+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	8.88	.833	YES
3+	.777	.66	YES
4+	.66	.5	YES
5+	.55	.33	YES
6+	.444	.166	HELLS YES

 

5+ lethal and sustained hits with a 4+ to hit

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.75	.833	NO
3+	.66	.66	Do you feel lucky?
4+	.5833	.5	YES
5+	.5	.33	YES
6+	.416	.166	YES

 

And here's a bonus one. Twin-linked with a 3+ to hit (like boltstorm aggressors)

To wound	Chance to wound per attack	Chance to wound per already successful hit	Worth fishing?
2+	.981	97.2	YES
3+	.926	.88	YES
4+	.833	.75	YES
5+	.703	.55	YES
6+	.537	.305	YES

 

 

I thought about doing dev wounds, but the occasions where you have rerolls to wound on a dev wound weapon are few and far between. That said, it's most useful on targets that are wounded easily but would be getting a good save against a normal wound. Shooting at Saint Celestine with a gladiator reaper's main cannon would be a good example.

 

Anyway, I hope this has been useful. 

Edited December 23, 2023 by Paladin777

[Tacitus]

On 11/22/2023 at 6:15 PM, Paladin777 said:

 

 

I thought about doing dev wounds, but the occasions where you have rerolls to wound on a dev wound weapon are few and far between.

Isn't it pretty much once per game with your one turn OOM boost? 

[Paladin777]

That and the TL Gatling cannon on the gladiator reaper, which is exactly my point on why I didn't bother.  

[Karhedron]

TLAC on the Land Raider variants too.

[Paladin777]

1 hour ago, Karhedron said:

TLAC on the Land Raider variants too.

Stop, you're gonna make me feel like I've got to do the math! XD 

[Tacitus]

5 hours ago, Paladin777 said:

Stop, you're gonna make me feel like I've got to do the math! XD 

It wouldn't hurt, but I look at fishing for 6's a lot like Football Coaches look at taking points off the board i.e. kick a (successful) field goal from the 45 on 4th and 15.  Get a 5 yard penalty on the defence making it 4th and 10 from the 40 - or keep the successful fieldgoal.  You keep the Fieldgoal. 

[Paladin777]

Fine, I'll do the math on fishing for DW! XD

 

Spoiler

2+ to wound

Enemy's save	Chance to inflict wound with normal to-wound roll	Chance to wound (after save) per already successful non-crit wound	Worth fishing?
2+	.277	.166	YES
3+	.388	.33	YES
4+	.5	.5	Do you feel lucky?
5+	.611	.66	NO
6+	.722	.833	NO

3+ to wound

Enemy's save	Chance to inflict wound with normal to-wound roll (after saves)	Chance to wound (after save) per already successful non-crit wound	Worth fishing?
2+	.25	.166	YES
3+	.33	.33	Do you feel lucky?
4+	.4166	.5	NO
5+	.5	.66	NO
6+	.5833	.833	NO

4+ to wound

Enemy's save	Chance to inflict wound with normal to-wound roll (after saves)	Chance for enemy to save per already successful non-crit wound	Worth fishing?
2+	.22	.166	YES
3+	.277	.33	NO
4+		.5	NO
5+		.66	NO
6+		.833	NO

5+ to wound

Enemy's save	Chance to inflict wound with normal to-wound roll (after saves)	Chance to wound (after save) per already successful non-crit wound	Worth fishing?
2+	.1944	.166	YES
3+	.22	.33	NO
4+		.5	NO
5+		.66	NO
6+		.833	NO

 

So ultimately fishing for devastating wounds is worth it much less of the time than one might expect. The incomplete tables are where I didn't bother finishing because as the enemy's save gets worse, fishing becomes worse and worse.

 

It's really only worth it if the enemy would be getting a 2+ save, or if you're wounding on 2+ and they have a 3+ save. 

 

Other than that those two scenarios, DONT FISH FOR CRIT WOUNDS!

 

@Tacitus I suppose, but the problem with that logic is that a field goal is a halfway point between getting full points for a touch down and no points. There is no such halfway point when it comes to rolling any given die in the game of warhammer (other than actual damage rolls, but that's a different topic). The results are 100% binary. You either deal damage, or you don't. There is no middle ground. While the results of actual dice rolls can vary, the probability calculations for any given scenario are immutable. As such, it's always prudent to take the option with the highest probability of success.

Edited December 3, 2023 by Paladin777