Some Rainy Day Probability

So I thought I would end this dry spell of posting with a couple interesting pieces of probability, one for warhammer fantasy, and one for 40k.

I'll look at wizards miscasting, and the probability of miscasting given the number of dice rolled for a spell. Often people will throw 6 dice at a spell hoping it will miscast, but do you really know what the odds are when you make that roll? You will now! Hooray!

Luckily a lot of game probabilities are simple enough binomial distributions and as long as you know your chance of hitting, wounding, saving, etc you can determine the likelihood of getting any given number of successes (in Bernoulli statistics there are only successes and failures, which translates excellently to wargaming). Here we consider irresistible force to be a success. 

We know you can't roll two 6s on a single die, and gamers generally know their 2d6 probabilities, so its safe for me to assume everybody knows, with 2 dice, getting boxcars is 1/36 = 2.777...%. I want my formula to work in general,  however, so verifying this will be a good check.

So. To find the probability of rolling a pair of sixes I need to know that the odds of rolling a six on a single die is 1 in 6, simple enough. Instead of working through the complication of finding all the results on 6d6 that have 2 or more sixes, however, it is easier to find the odds of rolling 0 or 1 six, and subtracting that from 1 (or 100%). This works because the probability of every possible outcome summed must add to 1.

If I let x be the number of dice rolled, my formula is P(miscast) = 1 - [ (xC0)(1/6)^0(5/6)^x + (xC1)(1/6)(5/6)^(x-1) ].

I'll break it down. xC0 means x choose 0, the combination function. This tells me all the possible combinations of n succsses in N trials. In this way NCn = N!/(n!*(N-n)!). So xC0 will always be 1, but I thought I would include it to be completely rigorous. I now know how many ways I can choose no dice of 3. I multiply this by the probability that that die rolls a 6, to the power of how many 6s I want, in this case 0. Any number raised to the power 0 is 1, but again I want to be thorough. Then multiply this by the number of dice I want to not be 6s, all three. Do the same for rolling a single six and we get the odds of rolling at most one 6. Subtract this from 1 and we get the odds of rolling at least two 6s. Nice.

Trying it out for 2d6 we get P(miscast on 2 dice) = 1- [(2C0)(1/6)^0(5/6)^2 + (2C1)(1/6)(5/60 ]

                                                                         = 1 - 0.97

                                                                         = 0.02777... or 2.77%

So it works.

Here's the list:

Probability of irresistible/miscast with

1d6 = 0%

2d6 = 2.77%

3d6 = 7.41%

4d6 = 13.19%

5d6 = 19.62%

6d6 = 26.32%

So one in four throws of 6 dice at a spell result in an irresistible/miscast. Good to know I guess.

I also wanted to look at something that was bothering me. Charging in 40k is now random, 2d6. Sure we all know the average is 7, and can be pretty confident in getting off 6" charge. If the target is in cover, however, that is changed to 3d6 dropping the highest die. This one is harder, and I didn't actually know the result. There is a general rule for 3d6 drop the lowest/highest which is to modify the average by 2, up to 9 or down to 5, but I wasn't completely satisfied by this.

I couldn't come up with a simple formula, and had to resort to a counting game. Finding this handy-dandy table was the only thing that made this endeavor possible.

So I can count up the results with two 1s in, the results with a 1 and a 2, the results with either 1 and 3 or 2 and 2, and so on. Here it is.

#       %           %>

2      7.41        100

3      12.50      92.59

4      15.74      80.09

5      16.67      64.35

6      15.74      47.69

7      12.50      31.94

8      8.80        19.44

9      5.56        10.65

10    3.24        5.09

11    1.39        1.85

12    0.46        0.46

So, turns out the most common occurring result is in fact 5, and the average is 5.54. Keep your models in 5" and you can play the odds pretty consistently. I know I won't be risking many 6" charges into cover with this knowledge. 

Let me know what you think. I can do more of these, and produce plots and all sorts of fun things, but if it just comes off as douchy then fuck it.

Ancient Greek Army Command Structure - Some Issues

I am having some trouble assembling my army for Ancients. I'm playing the Greeks, specifically looking at the time period around or just after the Trojan war and the Mycenaean age. I have lots of hoplites with short spears (I'm using the earliest hoplites to try and keep the time periods as appropriate as possible given the miniatures that I have) that I have assembled to go in a phalanx formation.

The issue arises with the command groups. In the army bundle I received 3 command packs. Each contains one hero on foot, one hero mounted, a banner and 2 musicians.
All the research I've done concerning hoplites and the phalanx formation has had little mention of banners or musicians. It seems the Greeks would perform celebrations and sing war hymns before going into battle (with the exception of the Spartans, who did have musicians following alongside the phalanx). This means that the phalanxes were not actually accompanied by a musician, even though I have 6 musician models. I also haven't seen a single word about banners. I saw a picture of a phalanx with some dudes running along the side, one of whom may have had a banner, but it was hard to tell.

As I see it the phalanx command structure consisted of having hardened fighters in the front rank leading each column of men, with the superior fighter on the left front to lead the charge, and his second on the right to support the weak side.
This is followed by rows of standard hoplites, and a few rows back would be another set of veterans to keep structure in the ranks.
I can easily paint up certain hoplites to represent this structure, but have no idea what to do with the command models.

On top of that, I have three mounted heroes, but my army contains a single, very small unit of cavalry (not including the chariots of course).
I don't believe it reasonable to have runners on horses, considering the phalanxes move slow enough to have runners on foot.

I could build command bases which would operate independently, but don't really have an in game use for them. I could build runners on quarters with a banner and musician and field them beside my units. Alternatively I could just build them and put them into the units, throwing accuracy to the wind for the sake of gameplay, but then what rank should each component go in? Would the banners and musicians be at the back, front?

I'm hoping somebody could lend some advice here, be it from your own knowledge of the history or simply an opinion that helps make the game mechanics work as well as possible. Also, there's no comment limit, so feel free to start a conversation in the comments (just in general, not that my post is so awesome it deserves more comments).

Early Musings on the updated Eldar

        So the first thing I want to address is that when 6th Edition came out the Eldar were not completely fucked. In theory, everything in the Eldar army worked as it was supposed to: ultra-shooty tanks that can zip around faster than a horny house fly, ultra-stabby assault troops with protection from ranged attacks and excellent movement rules, extra zappy mind powers that used to be the envy of the 40k Universe. But it was an untenable list mainly because it did these things for way too many points, and given the disparity between 4th (when the book was written) and 6th edition rules, some of the Eldar special stuff like holofields was badly in need of an upgrade. Their troops, Dire Avengers being really the only reasonable choice (and that in minimum units to occupy falcons and such), were totally outclassed by the average Chaos Space marine, who thanks to the new rapid fire rules could shoot at full distance even on the move, and who now cost a single point more than the Avenger himself, and could purchase support weapons.
Then the new tau book came out, and for a few extra points every one of their tanks could be made into a fast skimmer with a 4+ jink save instead of the regular 5+ that everybody now had to make do. It seemed absurd to me that the tau could simply have a better save than even the most badass of Eldar vehicles, until the new book came out and had in it the exact same upgrade.

That's really a sentence that tells a whole story. What this book did was update the Eldar, not fundamentally change them. Theyve simply been put into the same 40k-Fuck-Yeah line of development that spawned the Dreadknight and the Trygon as well as the teleporting Cryptek death squads. Now the Eldar have a a pair of flyers and main-listable titan to go alongside the Tau's Riptide (although the Wraithknight is noticeably larger). They also have a new wargear section (sort of, but its Eldar, they never really had one anyways) but with a few cool trinkets that get me excited about every new list I make. A model on a jetbike with the Mantle of the Laughing God gets a 2+ rerollable jink save and hit and run, but loses his independent character status. Faolchu's Wing lets a character run up to 48", and reroll all cover saves until his next turn if he does.

Despite saying they havent changed the basics, the new Eldar have a couple tricks their opponents might not have seen before. Battle Focus is an ability that allows Eldar to either run then shoot or shoot then run in the shooting phase, at absolutely no penalty (although they still forgo assault). This is the type of rule which really impresses on both players the Firstborn's swiftness, their total manoeuvring superiority of which even under-competent Eldar generals can take advantage. This rule is very similar to the German stormtrooper moves in Flames of war, except that there is no test for it. Add to this rules like Herald of Victory which makes sure that Swooping Hawks don't scatter, or that Warp Spiders count as Jet pack infantry now (except that they can jump further in the movement phase at the risk of getting swallowed), and you have an army that can be exactly where it needs to be to deliver maximum death to the enemy.

Most of these aspect warriors were also reduced in cost, primarily because they cannot capture objectives. Dire Avengers, the only aspect warriors in the troops slot, actually increased by a single point, putting them on par with a Necron warrior or Chaos Marine. However, there is a very good reason for this. The eldar have gotten a firepower boost across the board - their basic shuriken weaponry has what amounts to rending, allowing them to properly shred enemy infantry as they were always supposed to do (it is interesting to note that when rending became a rule in 4th ed my brother and I always thought that it would apply perfectly to Eldar guns, given that they gave it to Assault cannons... the 4th ed codex is a relic of an age where they still tried to restrain themselves). Scatter Lasers, the favourite coaxial weapon of most eldar generals, finally has a coaxial special rule, allowing a vehicle to reroll to hit with one other weapon if the scattler-gun hits first. Monofilament weaponry, already at strength 6 at least, gets +1 if the target is particularly slow or dull-witted - tanks are included, making a Warp Spider one of the most economical tank destroyers in the game (though with a short range). Oh right, and it gets AP2 if you roll a 6 to wound. Pretty brutal.

Im looking forward to a few more games, so i can test out some varied lists. Ive played already against Orks and Necrons, choosing to assume victory in both cases, and cant wait to shred up some imperials soon. Tau and Eldar, given their similarity, make excellent opponents, and there are some memorable scores to settle against Molgrimmarr the annihilator.