**Hey Everyone,**

Eric Riha reporting for guest article duty!

Eric Riha reporting for guest article duty!

**A friend of mine sent me a link to Tom De Mayo's excellent "Tank on Tank Love" article and I enjoyed seeing the oft forgot FoW Math on display – so much so that I felt the community could use a follow-up article (or three). The concepts explored in Tom’s article are very important in FoW, and provide a great lead-in/opportunity to talk about two other key concepts in the world of FoW Statistics: Expected Value and Potential Value.**

**In this first article of the series, I'll cover what Expected Value is and how to calculate it. In future articles, I'll talk about how it relates to Potential Value and (the important part) how to translate both of them into applicable tactics on the FoW board. I'm also going to cover this from the very ground up, so bear with me if I start too slow for you!**

Expected Value is exactly what it sounds like it is - what you should expect to roll out of a set of dice. Since we know there are only 6 results possible for each die roll, we can predict the Expected Value or Result of an action based on a certain set of conditions.

For example, let's take a quick look at infantry saves. The only condition in this setup is that infantry save on a 3+. On your standard 6-sided die (which is all we use in FoW - no d8's allowed!), that means you pass a save on a die roll of 3, 4, 5 or 6 and fail on a 1 or 2. Statistically, this means your possible rolls result in a fail 2 out of 6 times (or 1/3rd) and a success 4 out of 6 times (or 2/3rds).

Individual Expected Values don't tell us much (an expected save of .66666~ doesn't really mean anything), but it allows us to make educated guesses regarding what to expect in a set of dice. For instance, if someone shoots at me and hits 3 infantry teams, I should Expect to roll 2 infantry saves and 1 failure.

Expected Value is exactly what it sounds like it is - what you should expect to roll out of a set of dice. Since we know there are only 6 results possible for each die roll, we can predict the Expected Value or Result of an action based on a certain set of conditions.

For example, let's take a quick look at infantry saves. The only condition in this setup is that infantry save on a 3+. On your standard 6-sided die (which is all we use in FoW - no d8's allowed!), that means you pass a save on a die roll of 3, 4, 5 or 6 and fail on a 1 or 2. Statistically, this means your possible rolls result in a fail 2 out of 6 times (or 1/3rd) and a success 4 out of 6 times (or 2/3rds).

Individual Expected Values don't tell us much (an expected save of .66666~ doesn't really mean anything), but it allows us to make educated guesses regarding what to expect in a set of dice. For instance, if someone shoots at me and hits 3 infantry teams, I should Expect to roll 2 infantry saves and 1 failure.

Now, most of you probably already figured that last part out, but things start to get more complicated as we add more and more conditions to our equation. And calculating Expected Values are important because they give us a roughly 'average' result. If I have 5 stationary Sherman 76's shooting at a platoon of 4 StuG G's, concealed and at long-range; how many StuG's should I expect to kill? Is moving to close range with reduced RoF, but lower To-Hit numbers and Armor saves better for me? It is here that we can see where these calculations pay off. I should also mention that these types of equations should all be worked out before each game - taking 10-20 minutes to figure out the exact odds in the middle of a round might negatively impact your Sportsmanship score!

So let's use that first situation as another example. What are the conditions the bound this scenario?

5 Stationary Sherman 76's = 10 Shots

Sherman 76 Main Gun = Anti-Tank 12, Firepower 3+

Confident, Vet StuG = 4+ to Hit, +1 for Concealed, +1 for Long-Range = 6 to Hit

StuG G = 7 Front Armour, +1 for Long-Range = 8 Front Armor

Now, let's walk through the steps in FoW from shot to killed tank:

First, we roll to hit. Based on the above conditions, we have a 6 to hit which means we will score a hit on the StuG 1 result out of 6 possible results.

Now, most of you probably already figured that last part out, but things start to get more complicated as we add more and more conditions to our equation. And calculating Expected Values are important because they give us a roughly 'average' result. If I have 5 stationary Sherman 76's shooting at a platoon of 4 StuG G's, concealed and at long-range; how many StuG's should I expect to kill? Is moving to close range with reduced RoF, but lower To-Hit numbers and Armor saves better for me? It is here that we can see where these calculations pay off. I should also mention that these types of equations should all be worked out before each game - taking 10-20 minutes to figure out the exact odds in the middle of a round might negatively impact your Sportsmanship score!

So let's use that first situation as another example. What are the conditions the bound this scenario?

5 Stationary Sherman 76's = 10 Shots

Sherman 76 Main Gun = Anti-Tank 12, Firepower 3+

Confident, Vet StuG = 4+ to Hit, +1 for Concealed, +1 for Long-Range = 6 to Hit

StuG G = 7 Front Armour, +1 for Long-Range = 8 Front Armor

Now, let's walk through the steps in FoW from shot to killed tank:

First, we roll to hit. Based on the above conditions, we have a 6 to hit which means we will score a hit on the StuG 1 result out of 6 possible results.

Next, out opponent rolls an armor save. Based on the above conditions, we have a 12 AT to an 8 FA. Since we are looking for kills, we will ignore Bailed Results (ties). [As a brief aside, it is always best to ignore bails in your calculations as you should ALWAYS expect your opponent to roll to remount his tank.] This means our shot will penetrate the armor of the StuG when our opponent rolls a 1, 2, or 3 - 3 results out of 6 possible results.

Next, out opponent rolls an armor save. Based on the above conditions, we have a 12 AT to an 8 FA. Since we are looking for kills, we will ignore Bailed Results (ties). [As a brief aside, it is always best to ignore bails in your calculations as you should ALWAYS expect your opponent to roll to remount his tank.] This means our shot will penetrate the armor of the StuG when our opponent rolls a 1, 2, or 3 - 3 results out of 6 possible results.

Lastly, we must successfully roll a Firepower test to destroy the StuG G after its armor is penetrated. With a 3+ Firepower, we know that 4 out of 6 possible results will destroy the tank.

Now that we've walked through the steps in the game, creating Expected Values for each step, we need to combine those values together to produce a singular result. We do this by multiplying the fractions created by each step.

Lastly, we must successfully roll a Firepower test to destroy the StuG G after its armor is penetrated. With a 3+ Firepower, we know that 4 out of 6 possible results will destroy the tank.

Now that we've walked through the steps in the game, creating Expected Values for each step, we need to combine those values together to produce a singular result. We do this by multiplying the fractions created by each step.

Now, multiplying fractions is actually easier than is sounds. Split the top numbers (numerators) and the bottom numbers (denominators) into separate equations. Multiply each set of numbers separately to get the numerator and denominator of the cumulative Expected Value. Now stick those last two numbers into your calculator to get your final result.

Now, multiplying fractions is actually easier than is sounds. Split the top numbers (numerators) and the bottom numbers (denominators) into separate equations. Multiply each set of numbers separately to get the numerator and denominator of the cumulative Expected Value. Now stick those last two numbers into your calculator to get your final result.

Based on our calculations, we should expect to kill .05556 StuG G's for every 76mm Sherman round. With 10 shots from 5 stationary Sherman 76's, we would expect to kill .5556 StuGs - or a little over "half a StuG". Terrible odds indeed! In 'on-table' terms, we should not expect to kill anything in that StuG G platoon.

Moving to close range (with the StuG's still Vet + Concealed) gives us the following equation:

So we would expect to kill .14815 StuG's for every 76mm round. At 5 shots, that gives us an Expected Value of .74 killed StuGs - better, but still not all that great.

Based on our calculations, we should expect to kill .05556 StuG G's for every 76mm Sherman round. With 10 shots from 5 stationary Sherman 76's, we would expect to kill .5556 StuGs - or a little over "half a StuG". Terrible odds indeed! In 'on-table' terms, we should not expect to kill anything in that StuG G platoon.

Moving to close range (with the StuG's still Vet + Concealed) gives us the following equation:

So we would expect to kill .14815 StuG's for every 76mm round. At 5 shots, that gives us an Expected Value of .74 killed StuGs - better, but still not all that great.

Since none of the options above result in a single dead StuG, we know that if the above situation pops up in one of our games - we probably shouldn't take either of them! However, we can see that moving to close range (-1 to hit, -1 to armor saves) usually makes up for a reduced RoF and can use this as a general rule going forward.

Since none of the options above result in a single dead StuG, we know that if the above situation pops up in one of our games - we probably shouldn't take either of them! However, we can see that moving to close range (-1 to hit, -1 to armor saves) usually makes up for a reduced RoF and can use this as a general rule going forward.

And that's what Expected Values are really for - building a set of 'general rules/guidelines' for maneuver on the FoW board. By calculating these scenarios ahead of time, we provide ourselves with an expected result for each of our actions. Most ‘real-world’ scenarios on the FoW board will never match exactly with what you work out 'in theory'; however, the general concept of each mock engagement remains the same a majority of the time.

Calculating Expected Values also helps us re-align our expectations with reality, since we all 'usually' remember about "that one time where that one 105mm battery took out, like, 5 Panthers and it was really awesome - and I can't understand why it's not working this time....."

And since there are SO MANY different combinations of the above scenario (Panthers vs IS-2's, Sherman 75's vs Panzer IV's, Cruiser Mk IIIa vs. Panzer IV F, Concealed vs. Open, etc.), it is best to work them out for your specific army list vs. your generally expected opponent(s).

It is also a good idea to see where the 'turning point' is - what does it take to make each scenario "worth it"? Do you need more Shermans to get that StuG kill? Do you need to wait until they move out of cover? How many shots do you need to get that 1st Expected dead StuG?

So try some on your own, and join me next time when I talk about our next topic: Potential Value!

Eric Riha

FoW Mathamagician

And that's what Expected Values are really for - building a set of 'general rules/guidelines' for maneuver on the FoW board. By calculating these scenarios ahead of time, we provide ourselves with an expected result for each of our actions. Most ‘real-world’ scenarios on the FoW board will never match exactly with what you work out 'in theory'; however, the general concept of each mock engagement remains the same a majority of the time.

Calculating Expected Values also helps us re-align our expectations with reality, since we all 'usually' remember about "that one time where that one 105mm battery took out, like, 5 Panthers and it was really awesome - and I can't understand why it's not working this time....."

And since there are SO MANY different combinations of the above scenario (Panthers vs IS-2's, Sherman 75's vs Panzer IV's, Cruiser Mk IIIa vs. Panzer IV F, Concealed vs. Open, etc.), it is best to work them out for your specific army list vs. your generally expected opponent(s).

It is also a good idea to see where the 'turning point' is - what does it take to make each scenario "worth it"? Do you need more Shermans to get that StuG kill? Do you need to wait until they move out of cover? How many shots do you need to get that 1st Expected dead StuG?

So try some on your own, and join me next time when I talk about our next topic: Potential Value!

Eric Riha

FoW Mathamagician

## 20 comments:

Thanks for this, it's just what I've been looking for, especially as maths is a mystery to me. Can't wait to read the rest of the articles.

Good stuff!

Just to add some more thoughts...

Stabilizer would be 10 shots with a +1 to hit. This brings the to hit part back to 1/6, but the expected/round result is now .074... What? It's the same? Interesting, eh? I expect Eric's next article will show the difference between using stabilizer and not... :)

The next turn is where the shorter range gets you some benefit (assuming you survive...). You get 10 shots at .148 or about 1.5 expected kills.

Keep it coming!

Yep, he dives into the Stabilizer next article! baby steps for the folks who need em :)

As a general rule, the Stabilizer is even money if you are choosing between hitting on a 5 or 6.

The stabilizer is absolutely superior if you're doing anything else, and there's no use using the stabilizer if you're hitting on a 6 anyway.

You're not calculating the correct probability of getting a kill across ten hits with the Shermans. You multiplied your probability on 1 by ten instead of doing the proper binomial calculation.

The probability of at least one kill with the Shermans vs the Stugs should be 43%. The probability of at least 2 kills would be 10%, and the probability of at least three kills is 1.5%. Even with a probability of 1.5% you have even odds of seeing 3 Stugs drop to a single volley over the course of 50 games at least once.

The math for computing these types of probabilities is involved, but here's a link to a handy calculator: http://stattrek.com/online-calculator/binomial.aspx

You really need to make these calculations from both points of view. Making the decision to close the distance needs to involve how many Shermans you are going to lose to the StuGs as you maneuver for a higher odds shot.

Hi CaulynDarr,

The purpose of Expected Value is to give you a rough calculation for tactical purposes. It will not give you the exact percentage chance to kill one or more StuG's (in this case).

The main reason for using EV's over Binomial Calculations in Flames of War is that it gives you a single value instead of a set of values, allowing you to compare results between disparate sets of information (eg. Standing RoF vs Moving RoF). The second reason is that, with practice, you can begin to calculate EV's on the fly - or at least approximate them.

If you want to work out Binomial Calculation on the fly, then go for it - this class is not for you :D

Article 3, Kris :D

But, you're giving the wrong impression about the odds. You say that you should kill half a stug, but that's not useful. If they see 3 of their stugs die when they where confident that not even one should die, you haven't really helped them. It's more useful to say that approximately half the time you will loose at least one.

With a smart phone and the link I provided, you can figure these values out on the fly.

You're not wrong Caul, but expected value is a pretty solid barometer when facing different situations.

"With a smart phone and the link I provided, you can figure these values out on the fly."

Nah, if I or my opponent is taking the time to crunch numbers in their phone to make decisions, the game is quickly going to have the fun sucked from it!

I disagree. I don't think a % chance to lose X number of StuG's, vehicles, teams, etc. is any more or less valid than losing 'half a StuG'.

Using Binomials there is a 1.5% chance they can lose 3 StuG's - but they're still going to be confident they won't lose any because the chance to lose 1 is only 43%. If that % chance is less than half, then I think most of us would be confident that our StuG's would survive the encounter unscathed (maybe throw in a bail out).

I do talk about Potential Value - the scenario where you would lose multiple StuG's - in the next article because it is a valid in that set of data.

Binomials are accurate, but the process of calculating the breaks down the more dice you add to a situation.

If I have 40 MG dice from 10 T34/85's, firing at Vet infantry in the open, I can say the expected value for kills is 40/6 or 6.67 stands of infantry.

Using Binomials, I have to "guess and check" what my percentage chance to kill X number of stands for each possibility. I have a 99.9% chance to kill at least 1 stand, a 99.3% chance to kill at least 2 stands, a 97.2% chance to kill at least 3 stands - and so on. I fail to see how this is a more useful measure of estimated success.

More accurate - absolutely. More useful, no.

Just another nudge on the odds.

British and Canadian Shermans using "Limited Indirect Fire" The long range shot changes to 11/36 * 3/6 * 4/6 or a base value of .102 up from .056

This is great. I appreciate the in-depth explanations of the actual math. Its been a long time since Ive used any of this stuff.

Eric,

Thanks for the numbers. I am a BIG fan of probabilities and I appreciate the quick review above. I am very excited to read the next several articles that you have ready to post - if only Steve would get off his butt and put them up! ;-)

As an aside, I am very much looking forward to introducing myself to you at the next opportunity - perhaps LW Nats?

I have decided there are two types of FoW folks that "understand" the probabilities - those that run numbers (like me) and those that play so much that they have the probabilities beat into them without their knowledge.

And that is the real learning from your article:

You can speed up your learning curve as a player if you just take some time and learn the probabilities.

Thanks again!

Allen

Just a friendly reminder.. the 'Comments' on posts are for quick questions and 'thanks!' type stuff. Remember no 'counter-articles' to the articles. If you want a long discussion, please head over the the forums and write pages and pages of point/counter-point til your little fingers bleed. Enjoy!

My brain hurts

Excellent article, and good Simpsons reference to boot!

I need to make sure Charlie Clay sees this ;-)

Eric - glad to see you are spreading the gospel of actually understanding the risks that you are taking!

Ed

Yep, I never count bails as a success as well. Too many Fearless tanks and Protected Ammo for that to be of any reliable use. But it can help force a platoon morale check if the platoon has already been downsized.

Flames of Math....or...Math of War. :)

Cheers

We called it Combat Math when I was still Active Duty abd we had to "wargame" different Courses of Action during operations planning. Do it enough times and it becomes like second nature. Thanks for this series...plan to put it to use REAL soon. >:-)

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