Battle Report - 30-Jun-2014 - Chain of Command

View from the British end

As I've commented a couple of times in the past, the problem with playing different sets of TFL rules that are in some cases only subtly different is that bits of the rules specific to one set have an alarming tendency to drop out of one's head with disuse. Either that or I'm getting.. um... what's the word.... in my old age.

So, yesterday's game of Chain of Command did require quite a bit of flicking through the rulebook :D I was umpiring, Carl taking a platoon of British against AndyM with a couple of sections of Germans defending a small Normandy hamlet. After the patrol phase, Andy had two jump-off points in the village, and a third out wide on the left in a small wood. Carl had two centrally in a larger wood, and the third out wide on the same flank.

And then the madness started :D

Carl rolled command dice. 1, 3, 5, 6, 6. One pip on the CoC dice, a couple of activations and his phases again...

He rolled 1, 3, 5, 6, 6.

He then proceeded to roll ANOTHER THREE sets of dice with two sixes in, for a total of six consecutive phases (and enough 5s that he has a Chain of Command dice).

For those who weren't paying attention a while ago, the odds on rolling exactly two 6s on five dice is 16.1%. The odds on doing that five times in a row is that to the power 5, which works out as 0.011%, or one chance in a bit over nine thousand.

By then, Carl had a Bren team not quite on Andy's jump point, but close enough to deny him it, and a rifle team within 4" of one of the other two (also denying him the use of it), just outside the right hand building in the village, six rounds of mortar smoke covering his potential advance up the centre, and Andy hadn't deployed a thing. Fine initiative from the British.

Andy finally got to roll some dice. 1, 3, 6, 6, 6.

Pause for slightly hysterical laughter. These are my dice, a set from a Battlefront Open Fire box I reserve for command dice rolls in CoC when I'm umpiring so players don't confuse them with their own. My dice are legendarily bad.

Andy deployed a section in the village square, moved them so they could see the British outside the building....

...and Carl, perhaps wisely, used his Chain of Command dice to have them duck inside.

No problem. Some desultory fire rattled off the woodwork. The turn ends, as Andy rolled three sixes, so all the mortar smoke dissipated. Andy's roll.

Three MORE sixes, a four and a one.

After the laughter had died down again (just over one chance in a thousand, before you ask!) Andy brought his second MG34 team on with the one, to cover the village and the Bren team out on the flank. With the four, he deployed his senior Big Man, ordered two grenades thrown into the building and then ordered the section to close assault.

The British lost, got badly shocked, the rather paltry remnants fled, and the Germans followed up into the building to get out of the Bren's line of fire.

Which was probably about where we might have been after about this number of phases. Just, perhaps, not in that order!

From there on, things proceeded a little more sanely, thank heavens. The Bren on the right flank got taken out by the MG34 before it could actually capture the German jump-off point, and the British mortar likewise, eventually, by some naggingly accurate rifle fire. There was a firefight on the hedge line on the right flank, till the German section decided discretion was the better part of valour and pulled back into the village from whence it had come. The British gave chase, and the final action was a very bloody close fight in the square, which the Germans won, despite being down to four men, mostly due to having an MG34 and being on the defensive.

Another great game. Man, my dice are weird.

Probability for Wargamers - some Dreadball thoughts

After yesterday's rather satisfying strike (nearly the full length of the pitch), I got to wondering:

Suppose you have one action with your (all stats 4) Striker, and need to basically run, pick up the ball, run, pass, catch the pass with another Striker and score on that action. What are the odds?

Let's step through it.

To pick up the ball you need two successes (4,5,6) on four dice (remembering that 6's explode)... as it's late and I'm lazy, I'm availing myself of some superb work by BalrogBond on the Mantic Forums which you may need to read.

From the above, P(2 successes) = 29.1%

Now we have the ball: we run and throw with the free action we just got. It'll be a three dice roll assuming we got to short range - +1 dice for being a Striker, and -1 for having run. How many successes?

• Well, we have a 12.5% (1/2 * 1/2 * 1/2) chance of none at all. 
• For each dice we have a 41.7% chance of one success, so that's 3 * (50% * 50% * 41.7%, or just a hair over 31.2% chance of one on three dice. 
• for two it's a little hairer, as we can do it either by:
• one success on each of two dice (3 * 50% * 41.7% * 41.7%) = 26% PLUS
• two successes on one dice (3 * 50% * 50% * 6.9%) = 5.2%
• making a total of 31.2%
• for three successes there are several ways,
• one success on each dice = 7.2%
• three successes on one dice, 3 different ways = 3 * 1.2% * 50% * 50% = 0.9%
• two on one, one on another, six different ways = 6 * 6.9% * 41.7% * 50% = 8.6%
• for a total of 16.7%
• four? (my brain is starting to hurt). 
• 4 on one dice = 0.2% * 50% * 50% = 0.05%
• 3 one one dice, one on another, six ways = 1.5% (look, just trust me!)
• 2 on two dice, 3 different ways = 0.7 %
• 2 on one, 1 on two, 3 different ways, = 1.1%
• for a total of just over 3.3%

Let's stop there, as we have a long tail of 5+ successes.

The Dreadball throw/catch rules say we get as many dice to catch as we made throwing successes. We NEED two successes to get the final throw. because we need to double to get the free action.

• Two successes on one dice? = 9.3%
• on two dice? the easy way is (1 - the odds on one or none) = 33%
• on three dice? 56.3%
• four? 77.8%

So the odds of making the catch are (31.2% * 9.3% + 31.2% * 33% + 16.7% * 56.3% + 3.3% * 77.8%) = a smidgeon over 25% not counting the 5+ successes odds.

And our final throw is a three dice throw (+1 Striker, -1 small target). We only need one success, so that's an 87.5% chance ( 1 - the odds of none).

OK. Let's tot those up.

Final odds of scoring = 29.1% (pickup) * 25.1% (pass + catch) * 87.5% (strike).

About 6.4%. Wow. My dice were on yesterday!

So. Here's the question. 

You have one coaching dice. Where do you use it?

An evening at the club, and Dreadball

...and one of the many reasons I like our club :D

Talk about variety - last night we had:

Warhammer Fantasy
Dropzone Commander (must check this out sometime)
Dreadball (ditto, I have a team to paint)
WAB (playtesting for Carve Out A Kingdom)
Battlegroup Kursk
Bolt Action
Dystopian Wars (I think this got cancelled due to one player not being able to make it)
Battlefleet Gothic

Not bad for a Monday - we were impressively full - about 25 people.

Me? I spectated on the WAB for a bit, talked a lot about modding Chain of Command for a Falklands campaign with James (isn't modern technology fantastic when you can order a set of rules online and have them sitting on your tablet/phone inside 10 mins of starting the discussion!), and learned the mechanics for Dreadball.

On the latter? I know I'm late to the party here, but... Very very nice system. Jake Thornton would appear to have a knack for game design and manipulation of probabilities (which as a probabilities geek, appeals to me). I might have to start a second series of articles to look at some of the neat tricks he uses. The core system is based on rolling a number of d6 to score successes against a target number: some rolls you need 1 success, some you need more than your opponent. Neatly, if you get twice as many successes as you need, you get something Good - usually a bonus activation for that figure.

Also, 6s 'explode': i.e., any 6s count as one success and reroll to see if you get another. If you get another 6, you roll again, and so on. This is great partly because it means nothing is ever completely impossible (just sometimes really unlikely!), and with good dice you can pull off some really neat plays.

A couple of other nice mechanics: you can 'stretch' a run by one hex by making one success on a skill roll. If you want to go a second hex, you need another roll with two successes, etc etc. The throw/catch mechanic is nice - you get as many dice to catch a pass as you make successes on the throw. (Something the Vikes QB last night could have learned from!).

To add to the fun, you get a pool of coaching dice that you can add to any roll, but when they're gone, they're gone. All in all, it makes for a very neat system, where you have a lot of freedom of action and whether to take risks, without feeling completely ruled by random fate. I look forward to a proper game when I get my team painted up!

Probability for Wargamers - series summary

I think I've covered most of the useful techniques for figuring out the odds on various things, so we'll bring the Probability for Wargamers series of posts to an end with a handy index post for the series, for those of you who missed some of the earlier ones. Hope you've enjoyed these - there may be more later, who knows!

Probability for Wargamers 1 - Hitting on a 3 - what are the odds of hitting on a 3+ on a d6? What about failing to hit?

Probability for Wargamers 2 - The Gambler's Fallacy - why dice don't have memory, despite what you might believe

Probability for Wargamers 3 - Loot that Farm - looting rolls in Dux Brit, and figuring out how many rolls you'll need on average to succeed half the time

Probability for Wargamers 4 - More on looting - figuring out your best strategy

Probability for Wargamers 5 - The Tea Break Card - what are the odds on a card being drawn before the Tea Break/Tiffin/etc card in TFL rulesets?

Probability for Wargamers 6 - rolling two dice - the probabilities of various outcomes when rolling 2d6

Probability for Wargamers 7 - the myth of averages and rolling 3 dice - why the average result is going to get you killed, 90% certainty, and the odds on getting various numbers on 3d6

Probability for wargamers 8 - +1 vs reroll - working out how various game designers' favourite roll bonuses affect the odds

Probability for wargamers 9 - d20 vs 3d6 - looking at the graphs

Probability for Wargamers 10 - roll 3, keep worst - another interesting dice mechanic that skews the odds on target numbers

Probability for Wargamers 11 - combinations and Chain of Command - combinations - the most important tool for figuring out probabilities

Probability for Wargamers 12 - Chain of Command revisited - using combinations and common sense to figure out the odds of doing various things in Chain of Command

Probability for Wargames 13 - more Chain of Command - more odds, and discovering how, when you think clearly about something, it becomes a lot easier to figure out. And a homework exercise!

Probability for Wargames 13 - more Chain of Command

Derek raised an interesting point re last time's article:

"For example you want to work out the "chance of activating a team with a leader present".

The team itself can be activated if you roll one or more 1s with 5D6.

You can activate a Junior Leader by rolling any combination of dice that add up to three. That's one or more 3s, three rolls of 1, or one or more 2s, plus one or more 1s.

If there's a Senior Leader hanging about you can also activate the team with any combination of dice that add up to four.

It's all horribly complicated."

Yes it is :D However, in the case I was raising, that of activating a team, adding a 1 and 2, or 3 1s to activate the Junior Leader makes no difference to getting the team activated, since you can't do that without rolling a 1. But it does improve the odds on (say) being able to rally off a point of shock first :D

However, as our final piece in this series, at least for now, let's look at the really complicated one:

What are the odds on not activating a section (i.e. on a 2) in the presence of its Junior Leader (on 3) and a Senior Leader (on a 4)?

A 2, 3, or 4 will do: the odds of 5 rolls. none of which contain a  2, 3, or 4 are 1/2 ^ 5, or one in 32, or roughly 3.1% (by now you should be able to figure out how we get there), so the odds of at least one 2, 3, or 4 are 96.9%. In fact, we can skip this bit, but let's note that number anyway.

The only other roll that will get us the unit activated is to roll two or more 1s that add up to 2 or more. (Any other result that adds up to 2, 3, or 4 will already contain a 2, 3 or 4, and have been counted in the above 96.9%).

So, we need the odds on rolling at least two 1s and no 2s,3s or 4s. Odds on rolling a 1 are 1/6, on a 5 or a 6 are 2 in 6 or 1/3.

Odds of rolling all 5's,6's = 1/3 ^ 5, which is 0.41%
Odds of rolling one 1 and the rest 5s & 6s = 1/3 * 1/3 * 1/3 * 1/3 * 1/6 * ₅C₁, or about 1%
Odds of rolling 2 1s ... = also 1% (I'm not showing my working :D)
Odds of rolling 3 1s ... = roughly 0.5%
Odds of rolling 4 1s ... = roughly 0.13%
Odds of rolling 5 1s ... = 0.01%

A quick sanity check - all those should add up to our outstanding 3.1% which they do, as near as damnit (which is why that earlier calculation was useful).

Of course, by the time you've got to here, like me you should have realised that the only roll that will fail to activate our unit is to roll all 5s and 6s and no more than one 1. Odds, around 1.41%.

If you're bored, try a similar calculation for the same without the Senior Leader :D

What are the odds on a extra phase if you have only 4 command dice? How about 6?

If you've followed this series this far, you should be able to work it out. [Hint, you'll need to work out the cumulative odds of rolling k sixes out of n, using _nC_k.] I'll leave it as an exercise for the eager reader: your prize will be a namecheck in the wrapup post for this series.

Probability for Wargamers 12 - Chain of Command revisited

So, if you remember from last time, we introduced the concept of combinations of N things taken K at a time, and used this to work out the probability of rolling (for example) a given number of ones on 5d6, which (as anyone who's played the game by now will realise) is useful to know in Chain of Command.

Reproducing our table from last time: the odds of rolling K of a given number on 5 dice:

0: 40.2%
1: 40.2%
2: 16.1%
3: 3.2%
4: 0.3%
5: 0.01%

Useful. Taking this, we can figure out a few handy probabilities:

Chance of getting two phases in a row? 
You need 2 or 3 sixes: so that's about 19.5%. Let's call that one chance in 5 for the sake of the next one.

Number of phases before you get two in a row? 
If you remember the article on looting rolls in Dux Brit, you should be able to figure this out.

Chance of not getting a second phase = roughly 4/5 each phase.
Chance of failing to get one twice in a row = 4/5 * 4/5 = 16/25 = 64%
Chance of failing to get one THREE times in a row = 64/125 = just over 51%

To get that below 10%, i.e. to be 90% confident of having had an extra phase (which if you remember, is what we discussed in a prior post as maybe being acceptable odds), you need to roll TEN times.  Moral? Don't bank on that extra phase, but welcome it when it shows up.

Chance of activating a team?
Obviously enough, you need to roll a 1. Odds of rolling at least 1 one = 1 - (odds of failing to roll any 1s). So just under 60%. Again? Less than our desireable odds, so don't bet the farm on it.

Chance of activating a team with a leader present?
Let's say it's a MG42 team with a Junior Leader. You need either a 1 or a 3, i.e. 1 - (odds of failing to roll any 1s or 3s).

We'll have to calculate this one: we have 4 chances of 6 on any one dice of NOT rolling a 1 or 3, and 5 dice, so that's (2/3 * 2/3 * 2/3 * 2/3 * 2/3), which is 13.2%, giving us odds of 86.8%. Which is probably close enough, and demonstrates why leaders are really useful in Chain of Command.

How long will it take me to get a Chain of Command dice?
For now let's leave out the odd results, and concentrate on rolling 5s. How many 5s, on average, will we roll on 5d6?

Let's calculate the expected value. We have a 40.2% chance of 0, a 40.2% chance of 1... and so one. The expected number is thus (0.402 * 0 + 0.402 * 1 + 0.161 * 2 + 0.032 * 3 + 0.003 * 4 + 0.0001 * 5), or 0.83 fives. To get the 6 fives we need will take on average 6/0.83 turns, or just over 7.

Number of phases before the turn ends?
For this we need three or more sixes. Odds of this on any one turn are about 3.5%. If you really want to know, this makes a turn 50% likely to end after 20 phases. If you want 90% certainty? SIXTY FIVE phases. Don't wait for it, you'll have about NINE Chain of Command dice by then on average :D

Probability for Wargamers 11 - combinations and Chain of Command

Oi bin finkin', like....

The Lardies' Chain of Command has an interesting activation mechanic, and I've been pondering for a while what the odds are on various things happening as a result. To start with, we need to figure out the  probability of rolling a given number a certain number of times on 5d6. So let's start with the easy case.

What's the probability of rolling NO 1s on 5d6?

Easy!

Chance of not rolling a 1 = 5/6.
Chance of rolling 5 'not 1's' = 5/6 * 5/6 * 5/6 * 5/6 * 5/6 = 3125/7776 = roughly 40%!

How about probability of rolling one 1 on 5d6?

Chance of rolling a 1 = 1/6
Chance of not rolling a 1 = 5/6
Chance of rolling a 1 then 4 x 'not 1' = 1/6 * 5/6 * 5/6 * 5/6 * 5/6 = 625/7726 = about 8.1%.

Except that there are actually 5 different ways we can do this: the first, second... etc etc dice can be the 1. As these are independent events, we can add them, giving us a probability of roughly 40%.

What about rolling two 1's?

Simple enough, that's 1/6 * 1/6 * 5/6 * 5/6 * 5/6 * the number of different ways we can do that...

Which is 125/7776 * ... erm...

Right...

So...

We have 5 ways we can place the first 1. And for each of those, we have 4 different places we can put the other 1. So that's 5x4 = 20...

...except that since we don't care about the order, a dice in position 1 followed by a dice in position 2 is exactly the same as the reverse, so in fact there are only 10. Which makes the odds of rolling exactly 2 ones 125/7776 * 10, or 16% or so.

It turns out this 'pick k from n items' thing has a name, and a formula, which is going to save our sanity working through this. What isn't going to save my sanity here is that neither of the conventional ways of writing it is easy on a blog: however, with a little HTML wizardry....

The number of combinations of n things taken k at a time,

_nC_k = (n * (n-1) ... * (n-k+1) ) / (k * (k-1) * (k-2) ...)

As you can see, for our example above, and for the rest...
₅C₂ = (5 x 4) / (2 x 1) = 10 - odds of 2 1's = 16.1%
₅C₃ = (5 x 4 x 3) / (3 x 2 x 1) = 10 - odds of 3 1's = 10 x 25 / 7776 = 3.2%
₅C₄ = (5 x 4 x 3 x 2) / (4 x 3 x 2 x 1) = 5 - odds of 4 1's = 5 x 5 / 7776 = 0.3%
and obviously
₅C₅ = 1 - odds of 5 1's = 1/7776 = 0.01%

I think that car's relatively safe, Rich :D

Now we've got that, we can apply some maths to various things in Chain of Command.

See you next time for that!

Probability for Wargamers 10 - roll 3, keep worst

Another popular game mechanic - roll M dice, keep the best or worst N. D&D players should be familiar with it, as roll 4 dice, keep 3 was one of the more popular ways of rolling stats. In this case, though, the discussion cropped up on the TFL list, in respect of artillery fire in They Couldn't Hit An Elephant, their set of divisional/corps level ACW rules.

The discussion (and someone from the list will no doubt correct me if I missed the main thrust of it) revolves around breaking up groups of guns into sections, and how unrealistically bad the resulting likely effects are. A four gun battery gets 2d6: the game mechanics for a two gun section (only getting 1d6) result in it being fundamentally ineffective against infantry in line.

[Aside: one preferred notation for 3d6 keep lowest 2 is 3d6l2, which I'll use from here on in.]

Let's look at the basic odds for the 'roll 3 hits, keep the worst two'. I don't think this is what TCHAE uses, but Napoleon at War is fond of similar mechanisms.

2d6: success on a 4: each dice has a 1/2 chance of a success, so:

• 1/4 chance of no successes
• 1/2 chance of one
• 1/4 chance of two

Expected number of successes = (1/4 x 0 + 1/2 x 1 + 1/4 x 2) = 1, probability of two successes 1/4,

1d6: successes on a 4: you have a 1/2 chance of one success, expected number of successes = 1/2, probability of two hits 0.

Fair enough. Now let's try 3d6 (you should have paid enough attention in the previous posts to work this one out!)

• 1/8 chance of 0
• 3/8 chance of 1
• 3/8 chance of 2
• 1/8 chance of 3

But what we actually want is to take the worst two, which becomes

• 1/2 chance of 0 (1/8 chance of 0 + 3/8 chance of 1 successs reducing to 0)
• 3/8 chance of 1 (2 successes reducing to 1)
• 1/8 chance of 2 (3 successes reducing to 2)

And our expected number of successes is 5/8 with an 1/8 chance of two. Interesting!

I'll leave the detail for success numbers of 5 and 6 to you.

More interestingly, perhaps - I suspect (not having seen the rules) that TCHAE actually uses a IABSM-style fire table, so in fact what we're interested in is a proper 3d6l2. I'll leave the detailed maths to you, but here's the table and graph of chance of rolling >= a given target on both 2d6 and 3d6l2.

Roll	>= on 2d6	>= on 3d6l2
2	100%	100%
3	97%	93%
4	92%	80%
5	83%	64%
6	72%	48%
7	58%	32%
8	42%	19%
9	28%	11%
10	17%	5%
11	8%	2%
12	3%	0%

(note, the probability of rolling 12 on 3d6l2 is actually just under 0.5%, hence rounding to 0%)

The interesting points - the 50% value moves from just over 7 to just under 6, but the odds on getting a 7 drop from 58% to almost half that - 32% - as the curve drops off quite steeply compared to the straight 2d6 roll.

blue = 2d6, green = 3d6l2

Probability for wargamers 9 - d20 vs 3d6

While driving in this morning, I was listening to the latest View From The Veranda podcast with Neil Shuck and Henry Hyde (yes, yes, pedants - by which I mean they were on the podcast, not in the car - not that it wouldn't enliven my commute!). One of the topics that came up was the Beyond The Gates Of Antares Kickstarter, and the fact that it uses d10, not d6, from which the discussing drifted (as VFTV often does) into designers' choice of dice for such things as Phil Hendry's Augustus to Aurelian.

Somewhere about then it started to click that this series of posts is as much, if not more, about understanding probabilities for game design as to allow you the wargamer to understand (and, heaven forbid, mini-max) the odds. And I got to thinking afresh about the issue I mentioned in my Principles of War battle report the other day, namely the d20 morale roll.

Essentially (and I may have this slightly wrong since I'm working from memory, but the core concept is about right), in PoW units have a strength, which is typically a number around 10-12. Combat casualties reduce this strength, and a morale test is basically taken by rolling strength + modifiers vs a d20, as follows:

Roll	Effect
1	always succeed
<= Str	succeed
<= 2 x Str	shaken
<= 3 x Str	retire shaken
> 3 x Str	rout
20	always fail, one row worse than what the roll would be otherwise

My issue with this is that because it's on a d20, it's more prone to extreme results compared to a 3d6 roll. Go back and read up on the odds with 3d6, and then let me demonstrate with a graph.

The chart represents the odds (up the side) of rolling greater than or equal to a target number (along the bottom) on both 3d6 (the green curve) and d20 (the blue line). Key things to note: 

• the 50% point is the same for both rolls;
• the 3d6 is a fairly smooth curve that makes the extremes harder.

Take a look at the graph a different way, and you'll see the latter more clearly.

For the stats and math heads amongst you, the green curve (the 3d6) is what's called a normal distribution, Gaussian distribution or bell curve. For games designers, it has the useful feature that extremes are rarer, compared to a linear distribution (the blue curve for the d20). Note that the lines cross at about 6 and 15.  Particularly, note there's a 1 in 20 chance, or 5% of an automatic fail, which seems to me to be a bit vicious on a strong unit.

So, what would happen if we swapped the morale roll for PoW to be 3d6, rather than d20, and for the sake of completeness made 3 an automatic pass and 18 an automatic fail?

Well. The key thing is that auto pass and auto fail become much harder - 0.45% rather than 5%. I think this might be a bit too hard. So - what if we make that 3 and 4 always pass, 17 and 18 always fail. The odds are then a hair under 2% (4 chances in 216) of automatic failure, which feels about right as a compromise, and if we were feeling vicious we could make an 18 two column shifts from the actual result.

The other thing that happens (and I'll leave you to work out the numbers if you're interested) is that weaker units become a bit more likely to be shaken, and stronger ones less so. For example, if you look at the first graph, a unit of strength 5 is more likely to roll over 5 on 3d6 than d20. Interestingly, it's less likely to roll more than 10 (and thus retire) and considerably less likely to roll more than 15 and rout. It does make it a bit harder to rout units, in other words. 

I'm not convinced this is perfect by any means, as it seems to make routing a bit harder for units that probably should rout. But it does remove that nasty 5% chance of a perfectly sound full strength unit retiring shaken, which I think is a bit unrealistically excessive. Students of the Napoleonic era may disagree. Perhaps one approach to this would be to also change the table so that instead of the steps being at x2, x3, they're at +4, +8??

Thoughts?

Probability for wargamers 8 - +1 vs reroll

It's one of the rule-writers' common ways of making weapons in Warhammer-like d6 systems different. Some weapons get +1 to hit, some get a reroll. Which is better though?

Interesting question, and with the amount of stuff we've worked out in previous posts, not that hard.

With a target number of four, for example:

• Your normal odds are 3 in 6, or 50%.
• With a +1, obviously, they become 4 in 6, or 66.7%.
• For the reroll, you have a 50% chance of succeeding on the first attempt, plus odds of (chance to miss * chance to hit a second time), i.e. 50% + (50% of 50%). Which makes 75%. Clearly the reroll's the better bet. 

I won't work out the details for other target numbers: here's the table:

	Normal	+1 to hit	Re-roll misses
3	66.67%	83.33%	88.89%
4	50.00%	66.67%	75.00%
5	33.33%	50.00%	55.56%
6	16.67%	33.33%	30.56%

OK, so it looks like the re-roll's a better bet unless your target is hard to hit.

Another favourite of designers is the '+1 strength' modifier on the wound roll. Let's look at that: typically, for WAB, equal STR means wounding on a 4. 

	Hit+Wound (normal)	Hit+Wound (norm/+1STR)	Hit+Wound (reroll)
3	33.33%	44.44%	44.44%
4	25.00%	33.33%	37.50%
5	16.67%	22.22%	27.78%
6	8.33%	11.11%	15.28%

Again, interesting, The re-roll's still better, but less so. 

Moral of the story? Given the choice? Take the weapon that gives the re-roll!

Probability for Wargamers 7 - the myth of averages and rolling 3 dice

OK - let's move on to rolling 3d6. The most common thing I find myself using this for is movement in IABSM and/or Dux Brit. First question, then: what's the average roll on 3d6?

Easy enough to work out - it's three times the average roll on 1d6. Given an unloaded d6 rolled enough times, you'd expect an even spread of each number, so the average roll would be (1+2+3+4+5+6) / 6, which works out at 21/6, or 3 1/2. Which is probably one of the first stats about dice a lot of people learn. Cool: so, the average of 2d6 is therefore 7, and for 3d6 it's 10 1/2.

Fantastic, I hear you say. So if I'm 10" away from that unit I want to charge in Dux Brit, on average I'll make it.

Well.... yyyyyeeeessss.....

But let's look at the odds in more detail. I'm not about to find enough dice to do a pretty picture, so you'll have to make do with a table. the construction of which is left as (that phrase again) an exercise for the statistically-inclined reader:

Roll	# ways	Odds	Cumulative	Odds of >=
3	1	0.46%	1	100.00%
4	3	1.39%	4	99.54%
5	6	2.78%	10	98.15%
6	10	4.63%	20	95.37%
7	15	6.94%	35	90.74%
8	21	9.72%	56	83.80%
9	25	11.57%	81	74.07%
10	27	12.50%	108	62.50%
11	27	12.50%	135	50.00%
12	25	11.57%	160	37.50%
13	21	9.72%	181	25.93%
14	15	6.94%	196	16.20%
15	10	4.63%	206	9.26%
16	6	2.78%	212	4.63%
17	3	1.39%	215	1.85%
18	1	0.46%	216	0.46%

Digest that table, and let's take an example. Supposing you command a platoon in IABSM that's just activated: Another platoon has poured fire on a German MG42 in a defensive position, which has already had its actions and taken a fair bit of shock from your supporting fire, and some elements of your platoon are poised just over 10" out of close assault range. Next turn, the MG team is probably going to get three chances to shoot at you if you fail the assault (one or more of its own card, a Big Man and the Bonus MG Fire card), and you may get two (the platoon, and the platoon's Big Man if he's still in range).

Do you charge in? C'mon - the average distance you're going to roll is less than you have to move...

...but the odds on rolling 11" or more is ONLY 50%. Which gives you a 50% chance of being stuck in the open in front of an MG42 (plus whatever else is supporting it) that has a better chance than you of activating first next turn.

Do you charge in?

What odds would you like before you're comfortable giving that order? 90%+?