Five Easy Math Stats For MESBG

Good morning gamers,

If you've followed this blog for any length of time, you probably know that Rythbyrt and I really like math facts. Whether it's computing your likelihood of winning/killing in the context of heroic actions, special strikes, or making full-blown spreadsheets that iterate through two sets of combatants beating on each other, we love figuring out new, unexpected ways to maximize your army's effectiveness by delving into mathematical probabilities.

Photo Credit: Stock photo?

But what about players who don't want to be math junkies? Are there some simple and valuable things you can store in your brain to give you a ballpark estimate of what to expect in the outcome of your games? Well, today we put the complicated math on the side (sort of) and focus on five very simple things you can remember that will help you gauge your expectations (and maybe learn a little math along the way). Let's dig in!

Math Fact #1: Rolling 3 dueling dice = 70% chance of getting a 5+

In MESBG, the most exciting part of any game hands-down is the Fight phase. Unlike magic (which might be able to be resisted, but might not) and shooting (which you get absolutely no say over - though you can make your opponent work harder), fighting involves you rolling dice and your opponent rolling dice - and just because you have the better units doesn't mean you'll roll better than him. This makes the opposed rolls when dueling really important.

So this leads to our first math-based principle to remember: try to roll at least three dice in a duel. Why? Because rolling three dice gives you a 70% chance (0.7037 probability) of rolling a 5 as your highest number. Unlike in the board game Risk, you only care about your highest die result - if you can get 6s in all your duels, you have the opportunity to kill lots of guys and also have a great chance of not taking a lot of damage. The more dice we throw, the more likely we are to get a natural 6 (which is awesome).

Mathematically, we can compute this probability as follows:

• We have a 33% chance/0.3333 probability of getting a 5 or a 6 (or a "5+") on one die, since two-of-six faces have a 5 or a 6 on them;
• Of the 67% of the time that we don't get a 5+ on our first die, we have a 33% chance/0.3333 probability of getting a 5+ on a second die (which adds an additional ~22% chance that we get a 5+, since 67% divided by 3 is roughly 22%), giving us a cumulative ~55% chance of getting a 5+ on two dice; and
• Of the 45% of the time that neither of the first two dice got a 5+, we have a 33% chance/0.3333 probability of getting a 5+ on a third die (which gives us an additional ~15% chance that we get a 5+, since 45% divided by 3 is 15%), giving us a cumulative ~70% chance of getting a 5+ on three dice.

While the discussion above is centered around the dueling roll, it also applies to wounding rolls (if you're trying to wound an enemy model on a 5+, having two guys in a fight gives you a ~55% chance of killing him - which is why being wounded on 5s seems really, really easy, even though there's only a 33% chance of getting a 5 on any one die). Wounding probabilities are in general more complicated, though, since a 3 Attack hero (70% chance of getting a 5-high in the dueling roll) might have charged 2 models and might need two 5s instead of just one . . . yeah, computing the likelihood of killing two models with three dice is not something we're going to cover today.

This principle of 70% isn't just applicable to dueling roll expectations - I've used 70% as a reasonable threshold for casting magic against an enemy model in our Mastering Magic series, since striving for roughly 3/4 (or 7/10) of your casts to be successful seemed like a decent hurdle to cross. In a dueling roll, you can't guarantee that you'll roll well, but shooting for 70% chances of getting a 5 or a 6 is pretty decent (especially if you're a hero with a Might point left and a good Fight Value). Getting there is surprisingly simple if you can support friendly models with pikes (though positioning them to not trap each other can be tricky), spear-supporters near a banner, or charging cavalry models near a banner (or a host of multi-attack model options). No matter how you get there, most armies can get three dice in a fight - so try to do that if you can. Traps help too, by the way.

Math Fact #2: Rolling 4-7 dice = 50-70% chance of a natural 6

Similar to the discussion above, if you want to hit the 70% threshold for getting a natural 6, you want to roll at least 7 dice. Let's walk through this one to be pedantic:

• We have a 17% chance of getting a natural 6 on our first die, since one-of-six faces has a 6 on it;
• Of the 83% of the time that we don't get a natural 6 on the first die, 17% of those times we'll get a 6 on the second die, giving us a cumulative ~31% chance on two dice;
• Of the 69% of the time that we don't get a natural 6 on the first two dice, 17% of those times we'll get a 6 on the third die, giving us a cumulative ~42% chance on three dice;
• Of the 58% of the time that we don't get a natural 6 on the first three dice, 17% of those times we'll get a 6 on the fourth die, giving us a cumulative ~52% chance on four dice (so if you can get 4 dice in several fights, you should get a natural 6 in about half of your duels);
• Of the 48% of the time that you don't get a natural 6 on the first four dice, 17% of those times we'll get a 6 on the fifth die, giving us a cumulative ~60% chance on five dice;
• Of the 40% of the time that you don't get a natural 6 on the first five dice, 17% of those times we'll get a 6 on the sixth die, giving us a cumulative ~67% chance on six dice (which is interesting, since you'd think the answer would be "100%" since we're rolling 6 dice and each has a 6 on it . . . but that's not how math works); and
• Last one: of the 33% of the time that you don't get a natural 6 on the first six dice, 17% of those times we'll get a 6 on the seventh die, giving us a cumulative ~72% chance on seven dice (which breaks the 70% barrier, but just barely).

Now getting 7 dice in a fight is actually quite hard, but some models can do it (like the Watcher in the Water when charging in a water feature - yaye Monstrous Charge), but you can also pull it off if you have two big heroes or one big hero with a helper or two and maybe some spear supports and maybe a banner. Piling in warriors can also get you there, but usually this is after you've already torn through the enemy ranks and are doing "clean up" (or you're attacking something big, like a Dragon or a Great Beast of Gorgoroth). Isengard pike blocks can do it too if you're using Berserkers in the front rank and have two supporting pikemen and a banner nearby - but hopefully you catch the drift: rolling 7 dice isn't that hard to reach if you want it.

While rolling a 6-high is valuable for the dueling roll (you can't get any higher), there's value in knowing this for the wounding roll as well. If you're a S3 warrior and you're fighting a D6-7 model (warrior or hero - both are common), you'll need 6s to wound. If you're up against a lower Defense hero who has called Heroic Defense (like Galadriel or Gorulf Ironskin), you'll need 6s as well (and natural 6s at that). As was mentioned above, though, the wounding roll is a lot more complicated since you might need to do multiple wounds. For those interested in how many 6s you can get on 7 dice, I've dropped a snapshot from the builder I made that computes that (not going to explain it here - already done that):

Apparently, 7 dice have a ~10% chance of dealing 3 wounds when you need to roll 6s . . . that's not bad. I didn't display the probability of getting at least 5/6/7 Wounds on 7 dice because they were incredibly small (as you can see, you have less than a 2% chance of dealing 4+ Wounds). The probability of dealing 1 wound is the exact same score we got when determining our likelihood of getting at least one 6 above - funny how that works.

Math Fact #3: Roll 2/4/8 dice when you have the lower fight value

We've all been there - we've had an army that we like (maybe from Mordor or Angmar) and we look on the other side of the board and we see Elves. Lots of Elves. So many F5 (or F6!) Elves. And then we look at our own F3 Orcs (or whatever your fancy is - just not something that's F5) and we say, "Well, I won't be winning any fights." Yeah, it feels depressing, right?

Well guess what: what if I told you that from a math perspective, there was an easy way to turn a solid chance of losing into a slightly-in-your-favor chance of winning? Would you believe me? It all comes down to out-dicing your opponent. Let's take a quick look at the following scenarios:

The Elf has a narrow advantage in the duel (58:42)

In the fight above, we have 1 Galadhrim Warrior (F5) fighting 1 Uruk-Hai Warrior (F4). Both models will roll 1 Attack die and because the Elf has the higher Fight Value, when you make a grid of all the possible dice combinations they can get (see a previous post for how we do that), the Galadhrim Warrior has a 58% chance of winning the fight (so if we had just one-on-one fights in a long line, we'd expect the Elves to take three-out-of-every-five fights). Not great odds for us if we're the Uruk-Hai.

Add another Uruk and the Elf is narrowly outmatched (42:58)

If we can get a second guy in the fight, though, the tables actually flip! When you compute all of the likelihoods that the Uruk-Hai get a 6-high, a 5-high, etc., they are more likely to get high numbers (75% of the time they'll get at least a 4, which is better than the 50% of the time the Elf does) and now the Uruks have a 58% chance of winning the fight over the Elf (much better). This rule is simple: if your opponent has the higher Fight Value, engage a single 1-Attack model with two models.

"Okay," I hear you say, "that's all fine and good, but what if there's an Elf spearman in the fight? What does it look like then?" Well, things get worse for the Uruks: because the Elves are more likely to get 6s than they were before (which will auto-win the fight for them), the Uruks drop to a 39% chance of winning the fight (the Elves are back to winning three-out-of-every-five fights), which isn't great. So, to deal with this, let's flesh out the battle lines a bit for context:

Bad photo planning on my part - pretend the Elves aren't in range of a banner. 2 Elf dice vs. 4 Uruk dice is a narrow split (47:53) 

In the highlighted combat, we've got 3 Uruks in range of a banner against 2 Elves - and just like when the Uruks had 2 dice to the Elf's 1 die, having 4 dice when the Elves have only 2 dice gives us the edge again (the Uruks rise to a 53% chance of winning the fight). I've used pikes and a banner in this example, but if you're clever with the charging that you do, you can double-up in at least a few fights with spear-supports of your own, which will give you the requisite 4 dice you want. The principle here is clear too: if your opponent has the higher Fight Value, engage a 2-Attack model (or 1-Attack spear-supported) with 4 models (or at least get to 4 dice).

"But why can't the Elves have a banner," you say, "why do you hate Elves so much?!?!?!" I don't - really, I don't. So to be fair to them, let's see what happens when we give them a banner:

Good photo planning on my part - these Elves are definitely in range of a banner - and have a strong chance of winning (65:35)

If the fight were 3 dice vs. 3 dice, the Elves have a distinct advantage (they're getting a 5-high 70% of the time, which isn't great for the Uruks) and will win 65% of the fights (ouch). While we've been doubling the dice the enemy's facing in the previous two examples (extra credit point if you caught that on your own), doubling here doesn't get us to even odds. In fact, for the Uruks to have a 50% or better chance of winning the fight, they actually need EIGHT dice in the fight against the 3 dice rolled by the Elves to have a 51% chance of winning. That's . . . not great . . . at all.

But this also explains a lot about the game - why are Elf battle lines hard to crack through when they've got a supporting rank of spears and a banner? Why are banners so valuable to armies that have spears (and relatively high Fight Values)? Why are Fight 6/3 Attack heroes so hard to beat? It's probably because you're not rolling 8 dice when you fight them . . . and even if you are, it's still a 50/50 proposition for who wins the duel (or pretty close to).

If you engage a two-rank line of Elves (with a banner) with a two-rank line of your own that has a lower FV (with 3-4 dice in the fight on your side), you should only win one-in-every-three fights - and if you only have a 30-50% chance of wounding, you are likely to only kill 1-2 models out of every ten in the line (and with you losing ~7 fights, they're killing twice that from you).

So, if you find that you're out-Fight-Valued, don't lose heart - if you can get around the enemy flanks and charge the guys on the end, chances are good you'll not only get 2 guys into 1 of his models (which is sufficient to turn the dueling tables around), but you can probably angle your battle line charges a little so that you can get a few 4-on-2 fights. In all of these, you go from having a distinct disadvantage to slight advantage - and that can make all the difference.

Math Fact #4: Winning priority when you didn't have priority is roughly a three-in-five chance (58%)

Okay, so this one is much easier to see than anything we've already covered, since each side has one die being rolled. Winning priority feels good because you get to set the tempo for everything, but how likely are you to win it (all things being equal)? Well, that's an easy table to generate:

As you can see, of the 36 boxes in the table, 15 has a higher Horizontal die value (H), 15 have a higher Vertical die value (V), and 6 of them are tied (T) (which will go to whoever didn't have priority last round). If you divide 15 by 36, you get ~42%, while dividing 21 by 36 gives you ~58% - and those are your odds for winning a fair round of priority. To oversimplify the math, you should win three-out-of-five priority rolls when you didn't have priority on the previous turn and win two-out-of-five rolls to get back-to-back priority.

Now anyone who has played the game will tell you that you can go for four or five rounds without winning priority - that happens. The goal here is not to predict whether or not you'll win, but rather to set an expectation - 60% and 40% are pretty close together (and 58% and 42% are slightly closer), so it's possible to have one person have a run of good luck.

Now of course, not all armies fight fair - some armies can reroll their priority die (armies with Balin or Durin, for example) and others can boost or reduce their priority roll (armies with Elrond). And of course, there are two kinds of armies that can "just take priority one round each game" (Saruman-led Isengard and the Pits of Dol Guldur LL). Ignoring the auto-win-priority guys, you could compute your likelihood of winning priority with these factors in play by changing up the likelihood that you get a particular value to reflect the fact that the priority die can be boosted or rerolled (which becomes a 6-high, 5-high, etc. calculation just like we did above). But that goes beyond the purviews of this article. :-)

Math Fact #5: Take 12-18 ranged weapons to get ~1 wound/turn

This is perhaps the least inspired math fact, but I feel like it needs to be said here because many players seem to have landed on the number "6" when it comes to token archer models for their armies. Archery is a powerful component of the game and can allow a host of inferior troops to tackle a more superior band - but what draws people to six models vs. twelve (or more)? Well, let's have a look and make some guesses as to why.

Theory #1: Because I have a ranged weapon that wounds on 6s, I need six models. This theory makes sense, but it ignores the fact that you have to roll To Hit first, and THEN roll to Wound (so you'll probably get half your hits on target). The Green Dragon Podcast has popularized the idea of having a few "honesty" bows, but the way I see it, you shouldn't expect much from a handful of guys. If you already knew this, you might then fall into Theory #2: 

Theory #2: If I hit on a 4+ and wound on a 6, I need twelve models. While this is sometimes right, most archery does better at fulfilling scenario objectives when moving and shooting (because most scenarios require you to get somewhere - and most boards will have blocking terrain that you will have to maneuver around). If you have a 3+ shoot value, you can expect to do alright with 12 shots/turn, provided they can all see a target. But for everyone else (and even those with a 3+ shoot value), I would recommend 18 archers if you can.

If you are moving and shooting, chances are you will be hitting on a 5+. If you do, having 18 guys will give you the volume to overcome the bad shoot value and difficult wounding value. Now I can hear a few protests out there - so let's address a few here:

• What about crossbows? They always hit on a 4+ (or better if you have Vrasku) and usually wound on a 5+? While this is the on-paper calculation of the effectiveness of a crossbow gun line, I can tell you that getting all those models to shoot in one turn is difficult with their move-and-shoot restriction. Some of your crossbows will probably be moving into position for the next turn - so take more than six.
• What about S3 archery that can wound on a 5+? Yeah, you could get away with fewer archers because the Wounding difficulty is lower, but you again might need to move and shoot - and some archers may do better making a full move on one term instead of a partial move on two or more consecutive turns. 
• What if I can't get to 18? It's fine guys - if you won't/can't see why you need 18 ranged models or how you'd get that high in the first place, you should instead ask, "If I am unlikely to get a kill each round, what is my shooting actually doing for me?" Maybe you're okay with a kill every other turn - or maybe just a lucky kill before you're charged - that's fine, we're just setting expectations.

So yeah, shoot for 18 models (see what I did there?) if you can. Otherwise, don't get your hopes up. :-)

Conclusion

Well, if you're not a math buff, hopefully this post gave you a few things that you can keep in mind to manage your own expectations for your games (you'll win 3/5 priority rolls when you didn't just have priority and 2/5 priority rolls when you did just have priority, you want to have extra guys when you have a lower Fight Value, take lots of bows, etc.). These are statistical rules - obviously you could go five rounds without winning priority, kill someone with a single bowman, or win with a single Hobbit Militia in a fight. But hopefully you can at least keep in the back of your mind what "should" happen - and then not be disappointed when something you thought would work doesn't work. For those of you who are math-junkies . . . hopefully you got something out of this too. :) We'll be back with more math-based stuff in the future as well, but until next time, happy hobbying!