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Tuesday, February 7, 2017

"Never Tell Me the Odds:" How to Use Math to Make Better Gaming Decisions

While we all know whether something or not is a good idea in game by feel, sometimes choice A and choice B seem to be fairly even.

In Flames of War for example, by and large everyone knows that shooting twice at -1 to-hit with Stabilizers is always better than shooting once without, since if you get lucky you get TWO hits maximum and only 1 hit maximum without.. but what about something not as cut and dry?

Admittedly not a math guy like Eric Riha, I came across one or two cases where I really was stumped, and turned to some math for help.

I play Black Powder ruleset for my Napoleonics, and in that game, while every army has access to infantry, cavalry and artillery, there are a variety of formations that you can use, and my quandary was, should I bother moving my infantry up in Mixed Order (which means having skirmishers out front) and if so, it means I have to build and paint the skirmishing models!  They assault better in this formation, but I lose shooting power to do so.. which is actually better, and do I need to build more models, so I needed to be sure!

Here is my actual problem, and I'll go through math for my answer: Infantry can move up in a Line formation, where it can shoot 3 times w/ 18 inch range, hitting on 4+, enemy saving on a 4+, and a normal command bonus.   However, I can choose to set them up in an attack column because they are French, where they only shoot 1 time, shooting to-hit on a 3+, get a command bonus, but have -1 save.  Which is better vs similar stat'ed british line infantry?

Step 1, figure out the kill value for each set-up, and I'll use 3 units of British Line infantry and 3 units of French line infantry, no special rules (British have bonuses to line, French have even more bonuses when in Attack Column.. I'll ignore that for now)   In both cases, the French (on right) units move into shooting range (18 inches for muskets) and will be shot by the British (on left), as per the below pic:

Case A:  Move up in Line formation, shoot until the british are shaken. Moving does not hinder shooting, so 3 french units in Line formation get 3 dice each (9 total) hitting on 4+, and the British get a 4+ save.   Therefore, the chance of EACH SHOT scoring a wound is 1/6 + 1/6 + 1/6 (4, 5 or 6 to hit) times 1/6 + 1/6 + 1/6 (Brit player rolling a 1, 2 or 3 for saves) which can be simplified to 3/6 x 3/6 or 9/36 which is 0.25 per shot. Multiply by 9 since I have 3 units firing for 3 dice each, and we get an expected 2.25 wounds for the 9 shots taken.   It takes 3 wounds to 'shake' a unit, and if they take more than 3 wounds, they must take a break test.  Assuming they are shaken but did not break, the British fire back:

2 units firing exactly as above, so 6 shots x 0.25 chance to wound per shot is 1.5, however the Shaken unit is hitting on 5 or 6 only.  Their expected wounds is 1/6 +1/6 (5 or 6 to hit) x 1/6 +1/6 + 1/6 (me rolling a 1, 2 or 3 to save) which is 2/6 x 3/6 or 1/6 per shot.   we have 3 shots for the Shaken unit so 0.5 wounds.  1.5 wounds (unshaken units x2 firing) +0.5 wounds (shaken unit firing) nets us 2.0 wounds on average. Not enough to shake any one of our units.  We could now either keep firing or press home the attack and assault.   Below pic, we advanced 12 inches, took some hits, then focus fire (fences indicate the firing) on the center unit.

CASE B.   Move up in Attack Column within shooting range and fire at British infantry in a Line.   In Attack Column, I only get 1 shot per unit, but because I have the lead guys skirmishing, they are not shoulder-to-shoulder and get +1 to hit to shoot (all skirmishers get this rule as well)
Let's again calculate the chance of wounding per shot:   1/6 + 1/6 + 1/6 +1/6,  (3, 4, 5 or 6 to hit) x 1/6 +1/6 + 1/6 (if he rolls a 1, 2 or 3 to fail his save) = 4/6 x 3/6 = 12/36 = 0.3333 per shot.  Only 3 shots, so roughly 1 wound is expected.  Not enough to shake a unit, but will my formation now mitigate incoming damage?  Below, French are now in Attack Column with skirmishers out in front.

British fire back, are NOT SHAKEN as before, but are all hitting on a 5+ now because hitting skirmishers is -1 to hit, but my save will still be a 4+. British is in line formation as before, so will have 9 shots again.
Each shot will inflict: 1/6 +1/6 (hitting on 5 or 6) x 1/6 + 1/6 + 1/6 (me rolling a 1,2 or 3 to save) = 2/6 x 4/6 = 8/36 or  0.2222222 per shot.  9 shots means 2.0 wounds inflicted on me.

As you can see, in the first Case, I inflicted 2.25 wounds and took in return 2.0 back from 1 shaken and 2 unshaken units.. more if I did NOT shake a unit.   in Case B, I still took 2 wounds with no chance of shaking or breaking his units.   Without doing the math, I would ALWAYS run assault columns .. now I might want to consider advancing in Line and shoot him, soften him up, and THEN either charge as-is or gamble on passing the formation change test into Attack Column, then assaulting.

Now, one advantage of being in an assault column that is not shown in shooting stats is my command bonuses (I am more likely to take orders, such as Charging while in my Attack column) however I am even more vulnerable to cannon fire (attack column is tightly packed men, shoulder to shoulder, so I take more hits)   I still will assemble and paint my skirmishing companies for my Infantry Brigades, but will not automatically use them now that I know the math...

Lets move up a hundred years with a Flames of War real-life example, my first game vs Kurt Reese at a Late-War Nationals a few years' back had my 21st Panzer Div. vs his Grenadiers backed up by 2 Jagd-Tigers.

I had 3 self-propelled howitzers, the regular AT5, 2+ FP variety, and he was being extra-diligent in staying away from my observer.  In turn 3, he moves both Jagd's behind some 2-story houses, out of LOS from my Observer.. and then opts to run  infantry inside the house as well.. to be in bullet-proof cover.  My observer could see the house though!  With no other good targets, I ranged in on the Infantry first attempt, clipped both Jagd's with the template, killed 1 bailed the other and he failed the morale!!   We were joking around saying "What are the odds?"    Let's figure them out:

To range in:  I needed a 5 or 6 to range in, (infantry concealed in buildings, not gone to ground)   1/6 + 1/6 so 2/6 or 0.3333333

I then needed 3+ Skill to hit each Jagd (I honestly forget if I hit any of his dudes...) so that is 1/6 + 1/6 + 1/6 +1/6 (3,4,5 or 6 to hit) so 0.6666666 x 0.333333 = 0.2222222 chance of hitting each one after ranging in.

AT5 on top armor 2 means I score a bail or Penetrate if he rolls less than 3+, so 1/6 x 1/6 (1 or 2 to save) so 0.33333333 x 0.22222222 = 0.074 chance to kill, and 0.333333 x 0.3333333 chance to bail x 5/6 (FP 2+) is 0.093 for the bail.  Since I needed a Kill AND a Bail, we multiply together to get 0.006882 chance to force the morale.

Then he failed his 4+ morale save for the platoon (trained).. so the odds were less than 0.003441 or a third of a percent of all the above happening!!   1 in 300 chance, roughly.

Sorry Kurt!!  Good news is, in Version 4, my guns will have lower AT, per the WWPD V4 podcast!!

Obviously, you don't want to be doing this math during a game, because a punch in the teeth will be HIGHLY PROBABLE, but at least you can do a run-down of a case or two and see which gives you an edge.

Matt Varnish AKA Dennis Campbell, plays up in Canada with the lads from Surprisingly, I don't even have a Canadian army............ yet!    Check out some bat-reps on my youtube channel: 


Tom Mullane said...

nice Article! Thanks Matt!

John Gephart said...

Just the thing that I disliked about Avalon Hills General Magazine, turning wargames into a series of math problems figuring out the odds. It's the best way to suck the enjoyment out of a game.

Andrew Hopson said...

Good article. You explain the math very clearly for us non-math people. It also makes me want to buy a copy of "Blackpowder".

Rugi said...

Quote:"In Flames of War for example, by and large everyone knows that shooting twice at -1 to-hit with Stabilizers is always better than shooting once without, since if you get lucky you get TWO hits maximum and only 1 hit maximum without.. but what about something not as cut and dry? "

The above is actually incorrect when deciding between single 5+ and double shoots on 6s (in all other cases you should always take the double shoots and as there is not a single reason not to). In many cases it's a lot more important to score a single hit then scoring three, and shooting a single shot on a 5+ has a higher chance of securing a single hit than 2 shots on 6s.

The probability of hitting nothing on 6+ with 2D6 is 69%, while the probability of hitting nothing on 5+ with 1D6 is 67%!
So in cases when my one hit matters more (i.e. a hit at least bails the enemy which i can then assault and auto capture and maybe then even win, to reduce defensive fire with higher certainty), it's better to shoot a single time on 5+.

Mark G said...

Interesting how Black Powder favors not having skirmishers out front. That seems ahistorical, but I am just getting into the period with Grand Battles Napoleon (higher level up from BP).

Matt Varnish said...

Well, its all special rules aside, it doesn't favor having skirmishers up front. When you put in the French Attack Column rule in, it makes them easier to move around. @Rugi.. but then we have to get into more stats to see that while the probability of hitting nothing is slightly higher, you have a chance to get 2 hits.. so once you muck around with distributions, it IS better.. from a pure math perspective.

Rugi said...


On a long run 2 shots on 6s is not better than 1 on a 5+ at all. The expected number of hits in both cases is exactly the same.
So what matters is the current situation on the board. In case when getting 1 hit is all that you need, you are better off taking a single shot.
If the posibility of scoring twice could win you the game (i.e. two tanks to kill for the win) then of course you would go for 2 shots.
All in all, it just ain't true that taking 2 shots is always better. Expected number of hits is the same, what matters is the current situation and situational preference for the possibility of more hits with lower certainty of hitting anything or less hits with higer certainty of hitting at least something.

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