Heroic Strength: APPENDIX A: Heroic Strength vs. Mighting your to-Wound Rolls

I'm back! It's been a crazy month involving moving trucks and all that entails, plus trying to finalize my list for TMAT's upcoming Grand Tournament (I think I'm almost there... but I've thought that at least a dozen times in the last twelve days, so...). 

We closed out Part 3 of our series on Heroic Strength (you can find Part 1 and Part 2 here) by asking the following question:

Question 2: Does the probability of wounding with Heroic Strength outweigh the certainty of wounding if I save that Might point to modify a wound roll?

The reason we saved that discussion for later is that it really launches us fully into a secondary discussion of opportunity costs in general, which needed some additional problem-solving. I’m not a math expert, and the probability calculators that I usually use offer the probability of rolling a number combination on 2 or more dice (sum of 7, sum of 11, etc.), which is great for calculating courage tests (more on them in the future, perhaps) but not for providing a breakdown on the probability of rolling each individual number on X number of dice or the expected odds of rolling each number on each die, both of which are what we’re really concerned about here (in a D6-based game). Which means I had to get more creative.

Because I couldn’t find a single metric that adequately addressed this problem in a way I could understand, I’ve decided to tackle it instead using several different methods. In this Appendix, we’ll start with a batch of simulated dice rolls to give us a sense of what various to-wound rolls look like before and after Heroic Strength modifications. In a future write-up, we’ll consider some other, broader, methods, like expected outcome distributions and custom dice simulations.

Finally, if you're still wondering why we're still going on and on about Heroic Strength, which is clearly a niche heroic action, the data we're looking at is useful for other things as well--we're going to look specifically at wounding and the frequency with which +1 modifiers improve our dice rolls (and at what point we start suffering diminishing returns, or their opposite), so you can substitute your favorite +1 modifier for Heroic Strength/Might-to-wound (like two-handed weapons, Backstabbers, Hatred (X), etc.) or even +1 modifiers unrelated to wounding (i.e., +1 on climb tests for Cave Dwellers, or +1 on reinforcement rolls). Spoiler alert: +1 modifiers, especially +1 modifiers that operate on multiple dice, are tremendous value.

More maths awaits…

(which isn’t grammatically correct, but it alliterates… sorta…)

Approach #1: Absurdly Small Sample Sizes!

Approach #1 is the simplest to pull off, and by far the most imprecise: let’s get out some dice and just roll them!

In the spirit of full disclosure, I rolled these particular dice online (via roll-dice-online.com), so there’s always chance that Al Gore’s creation is somehow messing with our numbers. Setting that risk aside, let’s begin by rolling a single six-sided die (d6) ten times to get a baseline of 10 random rolls:

Roll #	Die 1
1	5
2	5
3	6
4	5
5	2
6	2
7	4
8	4
9	1
10	6

In and of themselves, this table tells us only what this single d6 happened to roll over the course of ten independent rolls. In other words, it's the most basic of basic dice data. Now let’s spice it up by adding some randomly-chosen colors and additional columns!

Successful To-wound Outcomes (6+):

Natural roll + 1 Might Point Spent (10 rolls on 1d6)

Roll #	Die 1	Wound    roll	+ Might    roll
1	5		1
2	5		1
3	6	1
4	5		1
5	2
6	2
7	4
8	4
9	1
10	6	1
	TOTAL	2	+ 3
	Avg/Rnd	0.200	0.300

We’ve now spruced up the chart to highlight the number of successful to-wound outcomes we’ve achieved on this single die, if we need a 6+ to wound (“6s”). Here’s the basic gist of what we’ve added:

·         We roll a single d6 ten times, and keep track of the individual rolls on that die, in the blue column (convenient labeled “Die 1”).

·         We’ve also counted the number of wounds (“6s”) we’ve rolled. I’ve marked successful wound rolls in the “Die 1” column in red (or salmon, or whatever), and have a separate column to the right that keeps track of how many natural to-wound rolls we’ve scored (in the column labeled “Wound roll”).

·         We’ve also singled out the number of “near-wounds” we’ve rolled (“5s”) that we can convert to wounds if we spend a single Might point: either by spending it directly to modify our to-wound roll (changing a “5” result to a “6” result), or by spending it on Heroic Strength (to improve our to-wound target from a “6” to a “5”). These are marked in yellow in the “Die 1” column, and are also tracked in the “+ Might roll” column at the far left.

·         Below each of the final two columns are some counters that track the total number of rolled wounds inflicted by this single d6 through 10 rounds (TOTAL “2”) as well as the total number of potential wounds we could add if we spent Might (TOTAL (“+3”).

·         Because these are cumulative totals over 10 rounds, I’ve also added a row beneath that, which breaks down the average number of roll/Might wounds we have in each individual round (AVG/Rnd “0.200” and “0.300,” respectively). What this tells us is that over the course of 10 rounds, this particular 1d6, which needed a “6+” to wound, rolled a 6+ an average of 0.200 times per round, and had an additional 0.300 “5s” on average each round that it could convert to a “6” if Might was spent.

Neither number is particularly impressive. If you’ve ever wondered why your High Elves have trouble wounding Uruk-Hai Warriors when locked in single combats… this is why. An elf who’s lucky enough to fight through ten rounds with the Uruks and rolls like we did would only deal 2 wounds over the course of those 10 rounds. Not exactly a killing machine. He could up those kills to a very impressive 5 kills per round if he had Might to spend… but he probably doesn’t (he does have only one attack, after all). The moral of the story (to the extent there is one): armies of roughly equal size who collide into one another, don’t have spears, and wound each other on 6s tend to devolve into grind fests.

Now let’s see what happens if we can improve our to-wound characteristic (through a 2H weapon, Piercing Strike, Heroic Strength, etc.). Now we’re wounding on 5s and 6s, and can also potentially wound on the roll of a 4 if we commit Might:

Successful To-wound Outcomes (5+):

Natural roll + 1 Might Point Spent (10 rolls on 1d6)

Roll #	Die 1	Wound    roll	+ Might    roll
1	5	1
2	5	1
3	6	1
4	5	1
5	2
6	2
7	4		1
8	4		1
9	1
10	6	1
	TOTAL	5	+ 2
	Avg/Rnd	0.500	0.200

First off, there’s more color, so right away we suspect something has improved. What we’ve done is converted all those yellow mighted-wounds from the last chart to basic wounds (reflecting that since we now wound on 5s, we no longer have to spend Might to wound on a 5). Just that change alone (going from a 6 to wound to a 5+ to wound) has saved this particular elf 3 Might over 10 rounds, even though he has no Might at all (don’t think about it too hard). Conversely, a S4 Uruk-Hai on the other side of the elven fight (who wounds High Elves on a 5+ instead of a 6) has managed to slay 5 elves over the course of 10 rounds, and could slay 2 more if he committed two might (which, again, he can’t, because he’s just a plain Uruk-Hai). Again, if you’ve ever wondered why Uruk-Hai seem to kill elves so quickly in one-on-one fights… this is a good chunk of the reason why.

Now we could keep going with this all the way to a 2+ (the last, reasonable to-wound score in the game), but with just a single attack die, it’s not particularly complicated or interesting, and also not representative of the models we’re actually concerned with here: heroes who have Might and Heroic Strength. So let’s add a second die and see what we can learn about two-attack heroes:

Successful To-wound Outcomes (6+):

Natural roll + 1 Might Point Spent (10 rolls on 2d6)

Roll #	Die 1	Die 2	Wound    roll	+ Might    roll
1	5	1	0	1
2	5	4	0	1
3	6	6	2	0
4	5	2	0	1
5	2	3	0	0
6	2	4	0	0
7	4	4	0	0
8	4	3	0	0
9	1	6	1	0
10	6	4	1	0
		TOTAL	4	+ 3
		Avg/Rnd	0.400	0.300

So you’ll notice that we’ve added a second column for a second attack die (conveniently labeled “Die 2”). Just to make sure we’re comparing apples to apples, I’ve kept all the dice rolls from Die 1 the same (and we’ll continue to do the same for Die 2, 3, 4, etc.) so that the only change we’re measuring comes from changing the number of dice, not their rolled outcomes.

The basic lesson from this chart is that wounding just on 6s sucks, even if we have 2 attack dice (which is why D6 troops against S3 troops, and D7 troops against S4 troops, feel so much more resilient than their D5 and D6 troop counterparts). In this particular example, our elf hero (who now has two attacks) improves his rolled wounds to 4 wounds over 10 rounds (compared to just 2 rolls with a single die), for a better-but-still-meager 0.400 wounds per round (or less than a wound in every two rounds). If we took a support hero this is perfectly fine, but if we took a combat hero (or at least what we thought was a combat hero), this is quite disappointing. Because we also rolled three “5s,” we could push our wounds up to 7 (instead of 4) if we’re willing to commit 3 Might. If that sounds familiar to you, it’s because we didn’t roll a single “5” on our second die. Dice are fickle, after all.

Now let’s say that our combat hero actually wounds on 5s instead of 4s (maybe because he’s an elf hero, and now has S4!).

 Successful To-wound Outcomes (5+):

Natural roll + 1 Might Point Spent (10 rolls on 2d6)

Roll #	Die 1	Die 2	Wound    roll	+ Might    roll
1	5	1	1	0
2	5	4	1	1
3	6	6	2	0
4	5	2	1	0
5	2	3	0	0
6	2	4	0	1
7	4	4	0	2
8	4	3	0	1
9	1	6	1	0
10	6	4	1	1
		TOTAL	7	+ 6
		Avg/Rnd	0.700	0.600

Just like our Uruk-Hai from earlier, all of our previously Mighted “5s” are now causing natural wounds, which is fantastic and pushes our total kill count for the game to 7 (without having to spend any Might! The paragon of efficiency!). This is what those probability tables we talked about at last time (remember, we jump from a 31% chance to cause 1+ wounds with 2 attacks needing “6s” to a 67% chance of causing 1+ wounds with 2 attacks if we can wound on a “5+”) look like when they manifest themselves on the table top.

In addition, we now have another 6 potential wounds we could score if we pour Might into them, which is both good and bad. Good because being able to wound a whopping 13 times over 10 rounds (or 1.3 wounds per round) is fantastic for a two-attack model—that’s more than a model per turn, on average. It’s bad because no two attack hero has anywhere near 6 might (unless Alfrid is lurking around your Lake Town captain, and Gandalf is lurking near Alfrid casting channeled Strengthen Wills… which some of you will spend the next few days thinking about way more than you should).

Now let’s pretend that our elf captain is actually an Uruk-Hai captain, and he has a two-handed weapon. Now he’s wounding on a 4+, instead of a 5+. Predictably, the result is a lot more red:

Successful To-wound Outcomes (4+):

Natural roll + 1 Might Point Spent (10 rolls on 2d6)

Roll #	Die 1	Die 2	Wound    roll	+ Might    roll
1	5	1	1	0
2	5	4	2	0
3	6	6	2	0
4	5	2	1	0
5	2	3	0	1
6	2	4	1	0
7	4	4	2	0
8	4	3	1	1
9	1	6	1	0
10	6	4	2	0
		TOTAL	13	+ 2
		Avg/Rnd	1.300	0.200

Now we actually are at the point where our little 2 Attack hero is doing 1.3 wounds per round on average. Additionally, he can now milk an additional two wounds out of these dice rolls if he spends two points of Might to change his “3s” to “4s” (which he can actually do, assuming he doesn’t spend any Might on other heroic actions, dice modifiers, etc.). And if our hero is actually able to get to the point where he wounds on a 3+, the damage potential nears two wounds per turn (which is fantastic for a model that only has two attacks):

Successful To-wound Outcomes (3+):

Natural roll + 1 Might Point Spent (10 rolls on 2d6)

Roll #	Die 1	Die 2	Wound    roll	+ Might    roll
1	5	1	1	0
2	5	4	2	0
3	6	6	2	0
4	5	2	1	1
5	2	3	1	1
6	2	4	1	1
7	4	4	2	0
8	4	3	2	0
9	1	6	1	0
10	6	4	2	0
		TOTAL	15	+ 3
		Avg/Rnd	1.500	0.300

Great, I can hear yourself saying to yourself over the interwebs, But I already knew Uruk-Hai were deadly and elves were sad. I thought this was supposed to help us evaluate Heroic Strength?

So here's how we're going to use these tables (of admittedly small sample sizes) to help us evaluate the value (or lack thereof) of calling Heroic Strength: to see if Heroic Strength is more efficient than spending a point of Might to change a roll, we check in each row and count the number of +Might wounds available to us (yellow boxes).

·         If there are no yellow boxes, then calling Heroic Strength would not have helped us at all (none of our rolls were high enough to benefit from a +1 modifier, or we already rolled high enough that we didn’t need a +1 modifier). In other words, keeping the Might in reserve is the more efficient choice.

·         If there is only one yellow box, then Heroic Strength and spending a point of Might to increase our to-wound roll would have had exactly the same impact on that particular roll. In other words, it’s a wash and neither choice helps us more or less than the other (on offense, at least).

·         If there is more than one yellow box, then Heroic Strength (assuming we get to the +1 modifer, which we will on a roll of a 3+ in every scenario, and could get it on a 1+ depending on our strength and the target’s defense) nets us value for every yellow box, while we’d have to spend an equal number of Might points for each yellow box to realize the same damage potential without Heroic Strength. In other words, Heroic Strength is the more efficient choice.

Looking back over our potential outcomes above, there was only one scenario (out of 40) where Heroic Strength proved more efficient than spending Might individually: Row 7 of our 5+ table, where our hero rolled double 4s and needed double 5s. In that limited scenario, Heroic Strength gets 2 wounds out of a single Might point. But that’s a rare case (just 1/36 rolls, as it turns out). Other than that, Strength managed a few draws, but keeping the Might in reserve for guaranteed damage proved the far more efficient use of our limited resources.

Now let’s scale it up to three attacks, and see if anything changes. This time, rather than reproducing our entire table four times, we’ll consolidate it into a single table, with color-coding and separated Wound/+Might totals differentiated by degree of difficulty (defense value). Like this:

Successful To-wound Outcomes Basted on Target Outcome (6+ to 3+):

Natural roll + 1 Might Point Spent (10 rolls on 3d6)

				Wound: 6+ (5+)		Wound: 5+ (4+)		Wound: 4+ (3+)		Wound: 3+ (2+)
Roll #	Die 1	Die 2	Die 3	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll
1	5	1	4	0	1	1	1	2	0	2	0
2	5	4	2	0	1	1	1	2	0	2	1
3	6	6	5	2	1	3	0	3	0	3	0
4	5	2	2	0	1	1	0	1	0	1	2
5	2	3	2	0	0	0	0	0	1	1	2
6	2	4	4	0	0	0	2	2	0	2	1
7	4	4	1	0	0	0	2	2	0	2	0
8	4	3	5	0	1	1	1	2	1	3	0
9	1	6	6	2	0	2	0	2	0	2	0
10	6	4	3	1	0	1	1	2	1	3	0
			TOTAL	5	+ 5	10	+ 8	18	+ 3	21	+ 6
			Avg/Rnd	0.500	0.500	1.000	0.800	1.800	0.300	2.100	0.600

So a couple of key things (as in the “key” in this new graph…):

·         We’ve gone from marking wounds and Mightable wounds in our “Die X” columns to color-coding them by degree of difficulty. Purple means we roll a wound on a 6, blue on a 5, green on a 4, yellow on a 3, and orange on a 2 (i.e., descending rainbow order). If we forget, the columns showing our Wound Roll / + Might Roll stats are color coded identically.

·         To figure out how many wounds we roll against a wound target (6+, 5+, 4+, 3+), we look at the color that corresponds to our wound target (for a 4+, for example, all wounds on a 4 are in green), and then add all rolls that are also marked in Blue (wounds on 5) and Purple (wounds on 6). To find how many dice per roll we could have Mighted up into wounds, we look for the color immediately below our target (so for a 4+ wound target, we Might up any yellow rolls). Or, if we’re feeling lazy, we look at the appropriate total columns on the right side of the chart.

·         The totals columns are identical to what we’ve seen so far, but have some additional color coding. The color we’re most interested in for this particular exercise is a solid yellow highlight in the + Might Roll column. This appears whenever a particular roll has 2 or more results that don’t count as wounds, but would if we expend Might. This is the magical circumstance where Heroic Strength becomes more efficient than just adding Might to completed wound rolls (i.e., it pushes all our to-wound dice up a damage tier, essentially allowing us to get an extra 2+ wounds out of a single point of Might, instead of 2+ Might). As you can see above, unlike our 2 attack die set (where this only occurred once), this occurs in four of our forty rolls (10%): twice when we were looking for 5s and rolled double-4s, and twice when we were looking for 3s and rolled double-2s. It’s still not an outcome that occurred the majority of the time, but adding one additional attack dice has increased the odds (as we now have greater odds than 1/36 of rolling at least two dice that have exactly the same number). Something to watch in the future…

·         The total columns also include solid red. This indicates that our roll has already inflicted the maximum number of possible wounds, which means no amount of Might will improve our roll. In the above scenario, three of our ten rolls manage to inflict their maximum number of wounds (3), although two of them need us to get to the point where we’re wounding on a 3+.

·         We also have red and yellow faded. These indicate that we’ve reached the point where we could inflict our maximum number of wounds if we expend 1 or more points of Might. This code immediately precedes solid red (since, if we move any lower down the to-wound table, we’ll score rolled hits).

·         Finally we have black. This is the inverse (sort of) of solid yellow, and therefore bad: if we spent a point of might on this roll, at this to-wound tier, we would have no dice that we could might up to a wound (with a single might point). Which means if we called Heroic Strength here, our Might point was wasted.

Based on this chart, even at three attack dice, our hero didn’t see consistent benefits from Heroic Strength at any point, although there were two consecutive rolls toward the end of the game where Heroic Strength would have netted our hero four additional wounds on just two Might. While that's cool, it's not consistent enough to prove an efficient use of Might.

At the same time, take a look at the total and per-round averages for how many mightable wounds are on the table. At 6+, adding that third attack dice scores 5 wounds over 10 rounds (or 0.5 wounds per turn, on average), which is noticeably higher (more than double) the 0.2 wounds we averaged with just 2 attack dice. And if we spent 5 points of Might over those same rounds, we could have had 10 wounds on a 6+, which is quite good. If we can get our 6+ to become a 5+, that’s worth 10 wounds over ten rounds (up from 7 with just two attack dice), plus we could have scored another 8 wounds off of Might boosts (for 18 total wounds over 10 rounds, because we rolled eight 4s over ten rounds), instead of the six bonus wounds (and 13 total wounds) available with just two attack dice. So keep an eye on those stats as we scale up further. Speaking of which…

Now let’s scale up to four attacks. This may sound high (Balrog / Sauron power-curve), but it’s actually quite common, either because we have a big-hero mounted on the charge against a cavalry model / monster, any basic 1-attack cavalry model who has charged and knocked prone infantry, or a basic 2 attack hero who is attempting to wound a model who is trapped or prone. Here are the stats:

Successful To-wound Outcomes Basted on Target Outcome (6+ to 3+):

Natural roll + 1 Might Point Spent (10 rolls on 4d6)

Roll #	Die 1	Die 2	Die 3	Die 4
1	5	1	4	3
2	5	4	2	1
3	6	6	5	5
4	5	2	2	4
5	2	3	2	5
6	2	4	4	5
7	4	4	1	4
8	4	3	5	5
9	1	6	6	1
10	6	4	3	3
	Wound: 6+ (5+)		Wound: 5+ (4+)		Wound: 4+ (3+)		Wound: 3+ (2+)
Roll #	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll
1	0	1	1	1	2	1	3	0
2	0	1	1	1	2	0	2	1
3	2	2	4	0	4	0	4	0
4	0	1	1	1	2	0	2	2
5	0	1	1	0	1	1	2	2
6	0	1	1	2	3	0	3	1
7	0	0	0	3	3	0	3	0
8	0	2	2	1	3	1	4	0
9	2	0	2	0	2	0	2	0
10	1	0	1	1	2	2	4	0
TOTAL	5	+ 9	14	+ 10	24	+ 5	29	+ 6
Avg/Rnd	0.500	0.900	1.400	1.000	2.400	0.500	2.900	0.600

Now we’re up to six of forty rolls where spending a single Might point to improve our wounding tier by one would net us two or more extra wounds. Granted, these are scattered across all four tiers, but it illustrates the central principle: the more dice we roll, the greater our chances of rolling at least two dice that are of the same value and are one less than the value we need to roll to wound. Notice as well that while our natural wound rolls for 6+ didn’t improve at all because we didn’t roll any 6s on Die 4 (what can I say? Some dice are just duds), our Mightable wounds have climbed to +9 (in other words, we could inflict 14 total wounds on a model that we require 6s to wound (i.e., a D7 or D8 model if we’re S4, a D8 or D9 model if we’re S5, etc.) if we had 9 might lying around and spent it.

We also have a whopping 3 extra to-wound rolls in our 5+ results on Die #7, because again, the more dice you roll, the greater chance we have of rolling two (or more) of the same value and just one less than what we need for a successful wound roll. And by the time we get to 4+ to wound, our hero is doing a whopping 24 wounds over 10 rounds (or 2.4 per round on four dice), without having to expend any Might. Hence, why the pointy sword of doom is so terrifying.

Onwards and upwards!

Successful To-wound Outcomes Basted on Target Outcome (6+ to 3+):

Natural roll + 1 Might Point Spent (10 rolls on 6d6)

Roll #	Die 1	Die 2	Die 3	Die 4	Die 5	Die 6
1	5	1	4	3	6	4
2	5	4	2	1	3	6
3	6	6	5	5	6	5
4	5	2	2	4	4	4
5	2	3	2	5	4	6
6	2	4	4	5	1	4
7	4	4	1	4	3	5
8	4	3	5	5	2	6
9	1	6	6	1	4	5
10	6	4	3	3	1	5
	Wound: 6+ (5+)		Wound: 5+ (4+)		Wound: 4+ (3+)		Wound: 3+ (2+)
Roll #	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll
1	1	1	2	2	4	1	5	0
2	1	1	2	1	3	1	4	1
3	3	3	6	0	6	0	6	0
4	0	1	1	3	4	0	4	2
5	1	1	2	1	3	1	4	2
6	0	1	1	3	4	0	4	1
7	0	1	1	3	4	1	5	0
8	1	2	3	1	4	1	5	1
9	2	1	3	1	4	0	4	0
10	1	1	2	1	3	2	5	0
TOTAL	10	+ 13	23	+ 16	39	+ 7	46	+ 7
Avg/Rnd	1.000	1.300	2.300	1.600	3.900	0.700	4.600	0.700

We’ve skipped five dice because it’s rare (and because I’m in a hurry to finish, as you probably are, too… we’re almost done). Six dice is not. Any big three-attack model will get there against a trapped or prone target, as well any generic mounted captain against prone infantry (and occasionally elite cavalry, like the sons of Eorl). Or the Watcher in the Water. Some thoughts:

·         First off, credit where credit is due: dice roll #3 remains incredibly hot (three 6s and three 5s, which would clearly be good enough for some kudo points at TMAT’s upcoming grand tournament). Die #6 is incredibly hot as well, rolling no worse than a “4” over all ten rounds.

·         We’re up to nine yellow boxes over ten rounds (40 trials), and are almost to the point where every roll has at least one (though Rolls 2, 5, and 9 still haven’t gotten there). Four of our nine would actually net us +3 wounds for a single point of Might (which would be fantastic value), and one of them (aforementioned Roll #3) nets us +3 wounds when we need 6+ to wound (which we’d be justifiably over the moon about).

·         Our + Might Rolls are also starting to differentiate themselves, with us seeing the greatest benefit at 6+ (we get 13 extra wounds if we can Might 5s into 6s) and at 5+ (an extra 16 wounds if we can convert 4s to 5s, including four of ten rolls where we’d improve at least two dice). The gains for 4+ and 3+ are less (+7 wounds each), which makes some sense (dice rolls that have already been improved into wounds at an earlier tier can’t be improved further later on), although we may be seeing some diminished returns. Something else to watch moving forward.

·         Our red streaks have also disappeared (except Roll #3!). It turns out getting max wounds on 6 dice (even if you can might 2s to wounds) is hard to do. Who knew?

Finally, look at our totals for 6+ and 5+ (i.e., vs. high defense models). Not only are they at a very solid average of 1.0+ wounds per turn (unmodified), but there are also (finally) no black squares. Which means every single roll would have gained at least +1 wound if we spent a point of Might (which means at worst, Heroic Strength would have gotten us an extra wound, which is as effective as if we’d held it in reserve). I won’t make you scroll back up to look at the other charts, but we’ve seen a steady decline in the number of rolls where spending a point of Might wouldn’t actually result in any more wounds (i.e., where declaring Heroic Strength was more likely than not wasteful):

	Wound: 6+			Wound: 5+
	No         + Might Wounds	+1 Might Wound	2+ Might Wounds	No        + Might Wounds	+1 Might Wound	2+ Might Wounds
1 Attack	7	3	0	8	2	0
2 Attacks	7	3	0	4	6	1
3 Attacks	5	5	0	4	4	2
4 Attacks	3	7	2	3	7	2
6 Attacks	0	8	2	0	6	4

The obvious caveat is small sample size, so let’s resist the temptation to read too much into this. Nevertheless, I think it’s safe to say that the more dice we add, the more our odds of at least drawing some benefit out of Heroic Strike improve (as do our odds of getting a major benefit), while our odds of “wasting” that Might point correspondingly drop.

One more…

Successful To-wound Outcomes Basted on Target Outcome (6+ to 3+):

Natural roll + 1 Might Point Spent (20 rolls on 8d6)

Roll #	Die 1	Die 2	Die 3	Die 4	Die 5	Die 6	Die 7	Die 8	Avg (Roll)
1	5	1	4	3	6	4	4	1	3.500
2	5	4	2	1	3	6	4	2	3.375
3	6	6	5	5	6	5	4	1	4.750
4	5	2	2	4	4	4	4	3	3.500
5	2	3	2	5	4	6	1	2	3.125
6	2	4	4	5	1	4	5	6	3.875
7	4	4	1	4	3	5	2	1	3.000
8	4	3	5	5	2	6	3	1	3.625
9	1	6	6	1	4	5	5	6	4.250
10	6	4	3	3	1	5	6	1	3.625
11	3	2	1	6	4	1	6	3	3.250
12	1	2	2	1	2	3	6	3	2.500
13	1	4	5	2	4	5	2	5	3.500
14	6	1	4	6	3	6	6	6	4.750
15	3	4	2	1	6	3	4	6	3.625
16	6	3	6	3	4	4	5	4	4.375
17	4	1	2	4	1	5	3	5	3.125
18	1	2	1	4	3	1	1	2	1.875
19	2	3	4	4	1	6	3	4	3.375
20	6	5	3	2	5	6	2	3	4.000
AVG (Die)	3.65	3.2	3.2	3.45	3.35	4.5	3.8	3.25
	Wound: 6+ (5+)		Wound: 5+ (4+)		Wound: 4+ (3+)		Wound: 3+ (2+)
Roll #	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll	Wound    roll	+ Might    roll
1	1	1	2	3	5	1	6	0
2	1	1	2	2	4	1	5	2
3	3	3	6	1	7	0	7	0
4	0	1	1	4	5	1	6	2
5	1	1	2	1	3	1	4	3
6	1	2	3	3	6	0	6	1
7	0	1	1	3	4	1	5	1
8	1	2	3	1	4	2	6	1
9	3	2	5	1	6	0	6	0
10	2	1	3	1	4	2	6	0
11	2	0	2	1	3	2	5	1
12	1	0	1	0	1	2	3	3
13	0	3	3	2	5	0	5	2
14	5	0	5	1	6	1	7	0
15	2	0	2	2	4	2	6	1
16	2	1	3	3	6	2	8	0
17	0	2	2	2	4	1	5	1
18	0	0	0	1	1	1	2	2
19	1	0	1	3	4	2	6	1
20	2	2	4	0	4	2	6	2
TOTAL	28	+ 23	51	+ 35	86	+ 24	110	+ 23
Avg/Rnd	1.400	1.150	2.550	1.750	4.300	1.200	5.500	1.150

I warned you about small sample sizes. ;-)

If we were tempted to conclude that at 6+ dice we’re guaranteed to have at least 1 die per round that we can change into a “6” or a “5,” we were mistaken. :-( The good news is that we did score just seven rolls in 20 tries where we would have wasted a Might point on either 6s to wound (6/20, giving us a 70% pass rate) or 5s to wound (2/20, for a very robust 90% pass rate). And even with that very poor stretch from rolls #11 through 19, we still managed to score an average of 2.3 mightable wounds per round, which means that (again, on average) spending a Might on Heroic Strength would have netted us an extra 1.15 wounds per attack if we needed 6s, putting us slightly ahead overall on the efficiency game. If we only needed 5s to wound, Heroic Strength would net us nearly 2 wounds per attack (+1.75), which is a very good return on just a single point of might, assuming we needed to deal more than the 2.5 wounds we were doing (on average) to the enemy per attack, with just a 10% fail rate.

While Die Roll #3 never managed to score that elusive 8th wound, Roll #16 did. And all eight dice ended up rolling slightly higher than average (somewhere above 3.2 per roll), with Die #6 rolling an impressive 4.5 on average through 20 rounds. Set him aside for those priority rolls you have to win!

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Again, we have to caveat this by mentioning that we’ve been looking exclusively at small sample sizes, so this is by no means a reliable measure of probability. At the same time, since our game is largely a game of small sample sizes (where 6-10 attacks is the norm), I found this approach helpful in helping me visualize exactly what Heroic Strength and a point of Might could do to a dice pool, before delving into more abstract statistical measures.

Next time, we’ll consider one more tool to help us measure the effectiveness of Heroic Strength (and lots of other things as well): expected outcome distributions.