This will be the last in our discussion of Heroic Strength. We left off last time in Appendix A looking at simulated dice rolls as a way of measuring the likelihood of "wasting" a point of Might on Heroic Strength. This time, we'll be tackling the same question using two more measuring tools: at least distributions and expected distributions.
An at least distribution chart attempts to convey the probability that we will score "at least" X results. For wounds (the primary thing we'll be mapping on these charts), this means our probability of scoring at least 0 wounds, 1 wound, 2 wounds, etc. out of however many to-wound dice our hero has. We made extensive use of these charts in Part 1, Part 2, and Part 3 of this series, looking at the odds that characters like Aragorn and Boromir would have of scoring at least X wounds when they wound on 6s, 5s, and 4s, so they're not entirely new to us.
An expected distribution chart measures the same data, but instead of telling us how many 0+, 1+, 2+, etc. we scored, this chart tells us the probability that we will roll exactly zero 6s, one 6, two 6s, three 6s, and so on. And since the probability of rolling a particular result (like a 6) is important for lots of reasons in our game beyond wounding, these charts may be of interest to you even if you've long lost interest in Heroic Strength.
(Maths ahead...)
Expanding Our Sample Size: Expected Distributions
To generate these charts quickly and easily (and because I'm not actually very proficient at math), I've opted to use a dice calculator. There are several out there that you can choose from, but for these tests I'm using anydice.com. This particular link will take you to a basic test (which is typed in the box that begins with "output") which calculates the probability of rolling a "6" ("count {6}") on two six-sided dice ("in 2d6"), which is a handy thing to know in our game.
The results, in percentages, are displayed directly below the box with our typed test, in the form of a bar graph. To the left of the bar graph is a "#" column which tells us the number result, and a "%" column which tells us the percentage of the time we scored that result. The default graph is an expected distribution graph (labeled as "normal"), but we can instantly shift to an "at least" chart by clicking the "at least" button (which we'll do in the next section; for now, we'll stick with expected distributions).
In the chart below, I've collected the results from four different queries and combined them into a single chart: if we have a single six-sided die (1d6), what is the expected distribution of rolling no "6s" and exactly one "6?" I then repeat the query for exactly no "6s or 5s" and exactly one "6 or 5" (5+), exactly no "6s, 5s, or 4s" and exactly one "6, 5, or 4" (4+), and finally exactly no "6s, 5s, 4s, or 3s" and exactly one "6, 5, 4, or 3" (3+).
Expected
distribution of Outcomes on 1d6
6s
|
5+
|
4+
|
3+
|
|
0
|
83.33
%
|
66.67
%
|
50.00
%
|
33.33
%
|
1
|
16.67
%
|
33.33
%
|
50.00
%
|
66.67
%
|
These numbers are pretty much fractional variants of 1/6, and not hard to understand: our odds of rolling exactly no sixes on a six-sided die is 83.33%, or 5-in-6, and our odds of rolling exactly one six is 16.67%, or 1-in-6. Our odds of rolling exactly one 4, 5, or 6 is 50% (or 3-in-6, which we can simplify down to 1-in-2), and our odds of rolling exactly no 4s, 5s, or 6s is also 50%. Again, nothing major here, other than that this gives us a good baseline from which to measure as we start adding additional dice (which we're also about to do). The orange denotes the most likely outcome, and is merely a visual aid to help us see where the most likely band out outcomes is.
Now let's see what happens when we add a second die. This time, we're graphing the odds of getting exactly "no 6s," exactly one "6," and exactly two "6s" (6s), exactly no "5s or 6s," exactly one "5 or 6," and exactly two "5s or 6s" (5+), and so on:
Expected
distribution of Outcomes on 2d6
6s
|
5+
|
4+
|
3+
|
|
0
|
69.44
%
|
44.44
%
|
25.00
%
|
11.11
%
|
1
|
27.78
%
|
44.44
%
|
50.00
%
|
44.44
%
|
2
|
2.78
%
|
11.11
%
|
25.00
%
|
44.44
%
|
We know intuitively that our odds of getting our desired result should increase by adding more dice, and that proves true here. At the same time, adding a second die alters the expected distributions of the precise results we'll see. Thus, as a random example, if we roll only a single die and need a 5+ to wound, our odds of rolling at least one 5 or 6 is just 33% (or 1-in-3), while our odds of rolling no 5s or 6s is a much larger 66.67% (or 2-in-3). That's a pretty significant fail rate. Adding that second die removes a sizeable chunk of those "fail" results, pushing our percentage of "0 result" outcomes from 66.67% to a less-than-majority 44.44%. Where does that extra 22.22(ish)% go? Half of it goes into improving our odds of getting exactly one successful result (pushing our odds from 33.33% to 44.44%) and the rest goes into a new probability: the chance to roll exactly two "5s" or "6s."
At the risk of stating the obvious, these distribution charts have numerous applications beyond our chance to wound. This could easily be our odds to win a duel (by going from one attack to two through shielding, a spear support, a banner reroll, etc.), casting or resisting a spell (adding a second Will point, or a reroll with Crown of Morgul), spending fate points to prevent a wound (or wounds), re-rolling climb/leap tests with something like Mountain Dweller, Rivendell Knights rerolling their to-hit rolls within 6" of Elrond, camel/chariot/Great Beast/Mumakil impact damage: basically anything we use 2+ dice for (or the equivalent of 2+ dice, when rerolls are factored in) and need a certain, specific dice result to accomplish. If our F5 elf will win the duel if he rolls a "6" due to his higher fight value, and he has to win the duel for some odd reason, he can push a good chunk of "fail" results (zero 6s) into "success" results simply by adding a second die (shielding, banner rerolls, or a spear support buddy). The same applies to a dwarf/Uruk fighting Orcs/generic men, Orcs/generic men fighting goblins/hobbits/ruffians, and goblins/hobbits/ruffians fighting hobbit militia.
Notice as well that the results for 5+ and 3+ have two outcomes ("0" and "1" results, and "1" and "2" results, respectively) which are both their highest expected results and yet neither is higher than each other. We'll see this repeated as we add more and more dice, so again, purely as a visual aid, I'll highlight the most likely distribution band in orange, with surrounding most likely outcomes in yellow, just to give us a sense of where most of the outcomes are expected to fall. And unsurprisingly, the more dice we add, the more our expected distributions get pushed around. Notice the progression:
Expected
distribution of Outcomes on 3d6
6s
|
5+
|
4+
|
3+
|
|
0
|
57.87
%
|
29.63
%
|
12.50
%
|
3.70
%
|
1
|
34.72
%
|
44.44
%
|
37.50
%
|
22.22
%
|
2
|
6.94
%
|
22.22
%
|
37.50
%
|
44.44
%
|
3
|
0.46
%
|
3.70
%
|
12.50
%
|
29.63
%
|
Expected
distribution of Outcomes on 4d6
6s
|
5+
|
4+
|
3+
|
|
0
|
48.23
%
|
19.75
%
|
6.25
%
|
1.23
%
|
1
|
38.58
%
|
39.51
%
|
25.00
%
|
9.88
%
|
2
|
11.57
%
|
29.63
%
|
37.50
%
|
29.63
%
|
3
|
1.54
%
|
9.88
%
|
25.00
%
|
39.51
%
|
4
|
0.08
%
|
1.23
%
|
6.25
%
|
19.75
%
|
Expected
distribution of Outcomes on 6d6
6s
|
5+
|
4+
|
3+
|
|
0
|
33.49
%
|
8.78
%
|
1.56
%
|
0.14
%
|
1
|
40.19
%
|
26.34
%
|
9.38
%
|
1.65
%
|
2
|
20.06
%
|
32.92
%
|
23.44
%
|
8.23
%
|
3
|
5.36
%
|
21.95
%
|
31.25
%
|
21.95
%
|
4
|
0.80
%
|
8.23
%
|
23.44
%
|
32.92
%
|
5
|
0.06
%
|
1.65
%
|
9.38
%
|
26.34
%
|
6
|
<
0.01 %
|
0.14
%
|
1.56
^
|
8.78
%
|
Expected
distribution of Outcomes on 8d6
6s
|
5+
|
4+
|
3+
|
|
0
|
23.26
%
|
3.90
%
|
0.39
%
|
0.02
%
|
1
|
37.21
%
|
15.61
%
|
3.13
%
|
0.24
%
|
2
|
26.05
%
|
27.31
%
|
10.94
%
|
1.71
%
|
3
|
10.42
%
|
27.31
%
|
21.88
%
|
6.83
%
|
4
|
2.60
%
|
17.07
%
|
27.34
%
|
17.07
%
|
5
|
0.42
%
|
6.83
%
|
21.88
%
|
27.31
%
|
6
|
0.04%
|
1.71
%
|
10.94
%
|
27.31
%
|
7
|
<
0.01 %
|
0.24
%
|
3.13
%
|
15.61
%
|
8
|
<
0.01 %
|
0.02
%
|
0.39
%
|
3.90
%
|
- The more dice we add, the further down the chart (i.e., skewing towards more results) we get in our probability band, especially as we move from left to right. Our three dice chart gives us a 34.72% chance of rolling exactly one "6," slightly more than double our initial 16.57% chance on just one die. We top out at a 40% chance of exactly one "6" if we roll 6 dice, before that number starts to go down. As it's doing so, more and more results are being siphoned out of our "exactly one 6" results into our "exactly two 6s" and "exactly three 6s" results, and as it does so, our band of most expected results shifts downward as well (towards more and more results). We see this at an extreme in our 3+ results, where we start with 66.67% of our results being exactly one 3+ on a single die, to more than 80% of our results being exactly four, five, six, or seven "3+"s by the time we have eight dice (again, watch out for mounted Aragorn two-handing the pointy sword of doom).
- Notice as well that for rolling "6s" in particular, our fail rate remains very high, even with an absurdly high eight dice pool (23.26%, or just shy of one-in-four). This has all sorts of implications: a charging F9 hero on horse who needs 6s to wound is going to roll exactly zero 6s almost a quarter of the time. But if he gets +1 to wound (from, say, an elven-made spear of magic), his odds of scoring zero wounds drops to just 4%, with the majority of his results likely to be exactly two or three wounds, and the most likely (i.e., expected) damage band being one, two, three, or four wounds. A three-attack hero with higher fight will win the fight automatically if he rolls a "6," but is likely to roll no 6s nearly 60% of the time (57.87%). But adding a mount, a banner, Lord of the West, etc. (pushing his dice pool to four, or effectively 4) reduces the odds of failing to 48.23% (or just shy of 1-in-2, which is still high, but better than 6-in-10).
- Expected distributions also provide a valuable counter-point to another common way of shorthanding probabilities: average outcomes. Suppose I have eight dice, and I need 4s to wound. On average, I should get a success in one of every two dice, which means if I roll eight dice, I should have four successes on average. This is in fact true, and as it happens, if you look at our eight dice chart you'll see that four successes is the most likely outcome (27.34% of our 4+ outcomes had exactly 4 results of a 4+). But the expected distribution chart also tells us that we have an almost equal chance of getting either 3 successes or 5 successes (21.88% chance of both) as we do of four successes (27.34%), that we have a better combined chance of getting either 3 or 5 successes as we do of getting four (43.76% vs. 27.34%), and that we also have a pretty good chance of getting either two successes or six successes (21.88% vs. 27.34%). If we happen to roll three straight results where we deal three wounds instead of four, it is true that our dice are "cooler" than average, but it is also true that our dice are rolling results that are entirely within the normal range of what we'd expect them to roll. In other words, "average" doesn't mean we're going to see that result every time, or even half the time, especially when we're dealing with small sample sizes (as we inevitably are in a turn-based and time-limited narrative game). An average is simply a measure of center, so having an additional measurement tool (like expected distribution) gives us a better idea of exactly how wide the center is, especially when we have a particular outcome we want to achieve.
Graphing more accurate probabilities: At Least Distributions
An at-least distribution chart basically takes all of the various probabilities that we have from our expected distribution chart, and squishes them together to give us the probability of rolling "at least" X of a given result. Take, for example, our basic 2d6 chart we looked at before:
Expected
distribution of Outcomes on 2d6
6s
|
5+
|
4+
|
3+
|
|
0
|
69.44
%
|
44.44
%
|
25.00
%
|
11.11
%
|
1
|
27.78
%
|
44.44
%
|
50.00
%
|
44.44
%
|
2
|
2.78
%
|
11.11
%
|
25.00
%
|
44.44
%
|
An at-least chart changes the expected outcomes from "exactly 0," "exactly 1," and "exactly 2," to "at least 0 (0+)," "at least 1 (1+)," and "at least 2 (2+)." Our odds of rolling at least no X results is always going to be 100%, because whether we roll an X result or not, we'll always have at least no X results (if that makes any sense). Our odds of at least 1 result is going to be the sum of our results with exactly 1 result, as well as any other results where we had 2+ results (since all of the 2+ results have at least 1 result in them). And we continue this until we reach the end of our results. So here, our expected distribution chart on 2d6 turns into this as an "at least" chart:
At
Least distribution of Outcomes on 2 attack dice
6s
|
5+
|
4+
|
3+
|
|
0+
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
1+
|
30.56
%
|
55.56
%
|
75.00
%
|
88.89
%
|
2+
|
2.78
%
|
11.11
%
|
25.00
%
|
44.44
%
|
Our chance of getting at least one 6 result is approximately 30%, which is... not great. If we need a 5+ instead, it goes up to ~56%, which is much better. Two dice will net at least one 4+ 75% of the time (again, Aragorn--and Banners!), and we'll get at least one 3+ almost 90% of the time.
Since we're no longer dealing with damage bands, I've swapped in some new colors. Green denotes that our odds of getting at least that result are over 50%, which means it's now more likely than not that it occurs (excepting our 100% baseline of getting no results, since we don't (usually) care about that probability, at least not when we're on offense). Note again that because average is a measure of center (and the center can contain quite a range of outcomes), counting on something that's a 50-50 roll is a dicey (sorry) proposition. For that reason, while there are almost always a few key moments in a game that come down to 50-50 rolls, we probably want to avoid wholescale tactics that rely on 50% probabilities (especially if we're throwing around Might, like we would be for Heroic Strength). Thus, I've arbitrarily chosen 75% or better as an ideal probability to build around, which I've marked in blue. Anything over 90% is about as close to a sure thing as we can have in a dice game, and thus will get the royal purple treatment (eventually).
Here's how the rest of the "at least" distribution charts look:
At
Least probability of Outcomes on 2 attack dice
6s
|
5+
|
4+
|
3+
|
|
0+
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
1+
|
30.56
%
|
55.56
%
|
75.00
%
|
88.89
%
|
2+
|
2.78
%
|
11.11
%
|
25.00
%
|
44.44
%
|
At
Least probability of Outcomes on 3 attack dice
6s
|
5+
|
4+
|
3+
|
|
0
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
1
|
42.13
%
|
70.37
%
|
87.50
%
|
96.30
%
|
2
|
7.41
%
|
25.93
%
|
50.00
%
|
74.07
%
|
3
|
0.46
%
|
3.70
%
|
12.50
%
|
29.63%
|
At
Least probability of Outcomes on 4 attack dice
6s
|
5+
|
4+
|
3+
|
|
0+
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
1+
|
51.77
%
|
80.25
%
|
93.75
%
|
98.77
%
|
2+
|
13.19
%
|
40.74
%
|
68.75
%
|
88.89
%
|
3+
|
1.62
%
|
11.11
%
|
31.25
%
|
59.26
%
|
4+
|
0.08
%
|
1.23
%
|
6.25
%
|
19.75
%
|
At
Least probability of Outcomes on 6 attack dice
6s
|
5+
|
4+
|
3+
|
|
0+
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
1+
|
66.51
%
|
91.22
%
|
98.44
%
|
99.86
%
|
2+
|
26.32
%
|
64.88
%
|
89.06
%
|
98.22
%
|
3+
|
6.23
%
|
31.96
%
|
65.63
%
|
89.99
%
|
4+
|
0.87
%
|
10.01
%
|
34.38
%
|
68.04
%
|
5+
|
0.07
%
|
1.78
%
|
10.94
%
|
35.12
%
|
6+
|
<0.01
%
|
0.14
%
|
1.56
%
|
8.78
%
|
At
Least probability of Outcomes on 8 attack dice
6s
|
5+
|
4+
|
3+
|
|
0+
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
1+
|
76.74
%
|
96.10
%
|
99.61
%
|
99.98
%
|
2+
|
39.53
%
|
80.49
%
|
96.48
%
|
99.74
%
|
3+
|
13.48
%
|
53.18
%
|
85.55
%
|
98.03
%
|
4+
|
3.07
%
|
25.86
%
|
63.67
%
|
91.21
%
|
5+
|
0.46
%
|
8.79
%
|
36.33
%
|
74.14
%
|
6+
|
0.04
%
|
1.97
%
|
14.45
%
|
46.82
%
|
7+
|
<0.01
%
|
0.26
%
|
3.52
%
|
19.51
%
|
8+
|
<0.01
%
|
0.02
%
|
0.39
%
|
3.90
%
|
The basic moral of the story is that as tough as needing 6s to wound is, if you continue to pile up dice, eventually you'll get there. That said, it's a lot easier if you can go from needing 6s to wound to 5s. :-P And if you only need 4s to wound, you have a pretty good chance of wounding (75%+) on as few as two dice.
Which brings us back to Heroic Strength (since I promised we'd talk about it specifically at some point).
Expected and At-Least Distributions: When is Heroic Strength a "Waste"?
In any heroic-resource-based decision we make, there are usually two outcomes: either the heroic action "works" and our decision was (probably) efficient (we win a heroic combat or we get to perform a heroic move first, assuming that we didn't call that heroic combat inefficiently to begin with, like calling a heroic combat where our hero doesn't want to move anywhere after winning and/or has no enemies to fight against after winning, or we call a heroic move after winning priority simply because we want to cross heroic move off our SBG bingo card), or it "doesn't" and our decision was (probably) inefficient (we lose the heroic combat or move roll-off, although there are situations where even a "wasted" heroic action can be efficient for us--if it forces an opponent to spend one of their last Might points, for example, or if it forces an opponent to not do something they were probably going to do anyway--like discouraging Thranduil from burning his one-time Nature's Wrath cast by calling Heroic Resolve).
Most of the time we know whether the heroic action was a waste or not at a particular point in time: either it worked or it didn't. And it either works once, or not at all. The "success" of Heroic Strength is slightly more nuanced, because it has multiple potential "successes" that can fire. Setting aside the potential defensive benefits (we may get a success if we prevent a knock-down or if we make a "rend" more difficult, and potentially we could get both successes), on offense we have the following potential outcomes:
- We need X to wound and get exactly no X-1 results--this is (probably) inefficient (although there are corner cases where we may have forced our opponent to spend Might on something else, like improving a duel roll or forcing a roll-off);
- We need X to wound and get exactly one X-1 results--this is potentially efficient, if we were already willing to commit at least one Might point to wounding our target (i.e., we got exactly the same result we were willing to spend Might anyway to achieve); and
- We need X to wound and get more than one X-1 results--this is definitely (probably) efficient, as we are gaining at least two wounds for a single Might point (the "probably" is if we're wounding only a single model with just a single wound...in which case it's not inefficient if we were willing to commit Might to killing that model, but only if we were willing to commit Might to killing that model... if that makes sense).
Both our expected and at-least distribution charts, in tandem, give us a pretty good idea at which of these three efficiency scenarios is most likely. For both charts, we're primarily interested in the column directly to the right of what we need to roll to wound (i.e., if we wound on a 5, we want our odds of rolling at least one 4+) and our rows of 1+ and 2+ results. Let's start with the basic two attack chart:
At
Least probability of Outcomes on 2 attack dice
6s
|
5+
|
4+
|
3+
|
|
0+
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
1+
|
30.56
%
|
55.56
%
|
75.00
%
|
88.89
%
|
2+
|
2.78
%
|
11.11
%
|
25.00
%
|
44.44
%
|
Here, we barely cross the 50% threshold if we need 6s to wound our target (a 55.56% chance of rolling at least one 5, which then becomes a 6). Our odds of at least one wound become quite a bit better if we need 5s (75.00% chance of 1+ 4s), and very good if we need 4s to wound (an 88.89% chance of at least one 3+). Having said that, this chart doesn't tell us (directly) how many of our results are "probably efficient" (one result) and how many are "definitely efficient" (2 results), so for that we break out the expected distribution chart:
Expected
distribution of Outcomes on 2 attack dice
6s
|
5+
|
4+
|
3+
|
|
0
|
69.44
%
|
44.44
%
|
25.00
%
|
11.11
%
|
1
|
27.78
%
|
44.44
%
|
50.00
%
|
44.44
%
|
2
|
2.78
%
|
11.11
%
|
25.00
%
|
44.44
%
|
But there's one more catch: the data above calculates our odds of rolling exactly (or at-least) a result of a 6, a 5+, a 4+, and a 3+. But Heroic Strength is only altering our dice rolls if we happen to roll exactly one or more results that are exactly one lower than we normally need to wound. So if we need 6s to wound, only rolls of a 5 will net us additional damage (as a roll of a 6 would have counted as a wound regardless of whether we called Heroic Strength or not). Similarly, if we originally needed 5s to wound, only rolls of a 4 will boost us, and only 3s if we originally needed 4s. So what we actually need is our odds of rolling exactly one or two dice with the "X-1" result on them. Where are those numbers?
Actually, they're right in front of us. The odds of rolling any given result on a six-sided die is the same regardless of the result (unless your die is loaded or cursed, of course): 1-in-6, or the same odds we have of rolling a "6" on our expected distribution chart. In this case, our odds of rolling exactly one six on two dice is 27.78%, or quite a bit less than 50%. Likewise, our odds of rolling exactly one "5" (if we call Heroic Strength and can boost 5s to 6s) is also 27.78%, as is our odds of rolling exactly one "4" and one "3." Which means on two attacks, the odds of us getting at least one mightable result is less than 50%, by a fair margin. And our odds of getting the hyper-efficient result of improving two dice for just a single point of Might is extremely small: 2.78% across the board.
Does this mean Heroic Strength is a waste of a might point in this scenario? Yes, and no, depending on how we quantify "waste." If we measure "waste" as "not being able to change one "fail" result to a "success" result (expected distribution chart), then yes; we only have a 27.78% chance of that occurring even once, which means it will happen less often than it doesn't. And paying that Might point for the aim of improving two dice rolls is a grossly inefficient choice (2.78%). But if our measure of success is not so much improving one or more dice rolls, but rather making our target easier to wound overall (at least distribution chart), I think there are definite places where that Might point is efficient. If we need 6s to wound, our chance of wounding in general (1+ wounds) improves from 30% to 55%, in other words, from "less-likely-than-not" to "more-likely-than-not." Notice as well that our odds of multiple wounds also goes up, from 2.78% to over 11%; against models with multiple wounds, that's not an insignificant boost, and it only improves the further down the chain we go (up to 44% if can wound by boosting 3s to 4s).
Now astute readers will note that we can get this same result, arguably with less risk, if we simply hold our Might in reserve and boost any results when they happen. Sure, we lose out on the potential double-bump, but as we just noted, that only happens 2.78% of the time (a far less likely benefit than the risk). And you'd be right. For two-attack models I'm not convinced Heroic Strength is worth it, because of that precise concern. But there are two caveats about that observation.
First, the "save it until you need it" argument works well at 2 attacks because we're accustomed to spending one might to boosting a single dice roll, and because the odds of rolling 2 or more of the same result are extremely small (less than 1-in-30). The argument loses steam the more dice we add, because as we add more dice, our odds of rolling exactly one and exactly two X-1 results grows significantly:
Expected
distribution of X-1 Outcomes on Yd6 dice
1d6
|
2d6
|
3d6
|
4d6
|
6d6
|
8d6
|
|
0
|
83.33
%
|
69.44
%
|
57.87
%
|
48.23
%
|
33.49
%
|
23.26
%
|
1
|
16.67
%
|
27.78
%
|
34.72
%
|
38.58
%
|
40.19
%
|
37.21
%
|
2
|
2.78
%
|
6.94
%
|
11.57
%
|
20.06
%
|
26.05
%
|
|
3
|
0.46
%
|
1.54
%
|
5.36
%
|
10.42
%
|
||
4
|
0.08
%
|
0.80
%
|
2.60
%
|
|||
5
|
0.06
%
|
0.42
%
|
||||
6
|
<
0.01 %
|
0.04%
|
||||
7
|
<
0.01 %
|
|||||
8
|
<
0.01 %
|
Which brings me to the second caveat: all the math so far has assumed that we only get to jump one damage band (i.e., from 6s to wound to a 5+ to wound). While this is the most likely outcome for most models (and not an inefficient option in its own right for heroes with 4+ wounding dice), there are scenarios where our hero will actually gain a +2 advantage to wounding if he rolls a 5 or 6 on the Heroic Strength die (i.e., S4 wounding D7 on 6s, becomes S7 wounding D7 on 4s). If that's the case, we know a couple of things:
- The hero is, at a minimum, going to get the odds we looked at above, because if they're in a position to gain a 2+ to wound, they're guaranteed to get a 1+ to wound even if they roll a "1" on the Heroic Strength test; and
- Their expected distribution chart now accounts for the odds of rolling two outcomes on a D6 (X-1, X-2), instead of just one, which means our expected and at least distribution charts now look like this:
Expected
distribution of X-1 + X-2 Outcomes on Yd6 dice
1d6
|
2d6
|
3d6
|
4d6
|
6d6
|
8d6
|
|
0
|
66.67
%
|
44.44
%
|
29.63
%
|
19.75
%
|
8.78
%
|
3.90
%
|
1
|
33.33
%
|
44.44
%
|
44.44
%
|
39.51
%
|
26.34
%
|
15.61
%
|
2
|
11.11
%
|
22.22
%
|
29.63
%
|
32.92
%
|
27.31
%
|
|
3
|
3.70
%
|
9.88
%
|
21.95
%
|
27.31
%
|
||
4
|
1.23
%
|
8.23
%
|
17.07
%
|
|||
5
|
1.65
%
|
6.83
%
|
||||
6
|
0.14
%
|
1.71
%
|
||||
7
|
0.24
%
|
|||||
8
|
0.02
%
|
At
Least probability of X-1 + X-2 Outcomes on Yd6 dice
1d6
|
2d6
|
3d6
|
4d6
|
6d6
|
8d6
|
|
0+
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
100.00
%
|
1+
|
33.33
%
|
55.56
%
|
70.37
%
|
80.25
%
|
91.22
%
|
96.10
%
|
2+
|
11.11
%
|
25.93
%
|
40.74
%
|
64.88
%
|
80.49
%
|
|
3+
|
3.70
%
|
11.11
%
|
31.96
%
|
53.18
%
|
||
4+
|
1.23
%
|
10.01
%
|
25.86
%
|
|||
5+
|
1.78
%
|
8.79
%
|
||||
6+
|
0.14
%
|
1.97
%
|
||||
7+
|
0.26
%
|
|||||
8+
|
0.02
%
|
With as few as three dice, we can have a 70% chance of getting at least one boosted die out of our Heroic Strength, and at 6 dice, our most likely outcome is two (or, for eight dice, three) boosted dice out of a single might point. In other words, it's very hard to "waste" it. Yes, we have only a 33% chance of realizing that bonus. But for infantry combat monsters there's very little risk involved (since we're getting at least a +1 to-wound bonus anyway), and for mega mounted heroes, I think I'm safe in saying there's no other way you'd be able to improve your hero's damage, to that degree or volume.
Especially if you stack channeled Enchanted Blades. ;-)
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