"Dice, Field of Glory & Hard Sums" Topic

Another blog article:

"I've had some feedback on the original article on dice, DBMM & FOG which in turn led to an even more interesting telephone discussion about whether rolling more dice in Field of Glory is better for you. At first glance questioning this seems stupid. "Of course it is" I hear you cry! Well I've had a look at this and the answer is both yes and no."

bit.ly/9crguF

Not for the faint hearted. Have fun!

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Martin Stephenson
vexillia.blogspot.com
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it's an interesting insight. I do certainly agree that extreme results become less likely as more dice are rolled, but only at the extremes.

Say, for example, you're rolling two dice. Your chance of 2 hits is only 25%, the balance of the probabilities being in the center at one hit.

Now, if you roll four dice, your chance of rolling two hits goes up dramatically -- to 37%.

So, it seems that, if you're looking for an intermediate result, more dice is better. But if you're hoping for maximum hits, the probability gets smaller as you roll more dice.

I tend to look at the same result, not the maximum. So, if I'm looking for two hits, I have a 25% chance with two dice, but a 69% chance with four, and and overwhelming 89% with six dice.

I do certainly agree that extreme results become less likely as more dice are rolled, but only at the extremes.

See bit.ly/9sAojM as is looks like Groucho got there before you. :-)

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Martin Stephenson
vexillia.blogspot.com
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As a person with some background in math (I did engineering) I appreciate what you are doing. However, when it comes to wargaming I purposely try to not get into the stochastic side of dice roles, even though sometimes it's hard to not think about it. The reason is I want to take the 'game' elements out of it and want to be play as if I were a general. When I get start calculating probabilities it breaks the immersion.

However, that's mean to take away from what you're doing, I like the blog and certainly the math that you are doing has merit.

Regards

Hi JJ

I agree wholeheartedly. I want to think like a general too! That's why I'm enjoying FOG more than DBMM at the moment.

I started looking at the maths because the way the dice work in FOG is different from DBM and much harder to estimate at a tactical level. The maths has shown me why and I thought the analysis would help others.

More importantly, it has created a framework for me. With it I hope to stop undermining my generalship with false tactical hopes and fears based on dodgy estimates of success or failure.

By the way I liked the phrase "When I start calculating probabilities it breaks the immersion.". Mind if I borrow it?

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Martin Stephenson
vexillia.blogspot.com
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Feel free to use the phrase.

Statistics. When you analyze 1 outcome, you are correct in that you will see more wild swings with smaller amounts of dice. When you test for the same outcome over and over again, your luck will tend to balance itself out as you are adding more and more dice to the equation. Example: you have 3 units of light horse trying to hold a line instead of just 1. That might not work out so well for you.

While it is good to know the statistics, the larger question should always be, "How much damage will be caused should I fail?"

Interesting blog!

John

When you test for the same outcome over and over again, your luck will tend to balance itself out as you are adding more and more dice to the equation.

Not quite. When you repeatedly test something the odds for each test remain the same whilst the rolling average of all the results combined approaches the statistical average as previous high and low scores counteract one another.

However you can't combine the rolls from different combats for three reasons; two maths and one FOG related.

[1] The FOG reason is that high and low scores coupled with failed cohesion tests for either side will change the number of dice between each round of combat. So you can't repeat the test (or combat) ad infinitum.

[2] The maths reason #1 is that, even if things stay the same, each round of combat is an independent event and previous results have no impact on its outcome. The average changes; not the odds for each combat.

[3] The maths reason #2 is that the approach to the average is governed by the "Law of Large Numbers". The numbers of dice involved in this far exceeds those rolled in one FOG combat. Have a play with "Law of Large Numbers" demo mentioned in the blog article to see this in action. Be sure to show the mean and roll totals; the latter continue to vary whilst the mean converges on the average.

Example: you have 3 units of light horse trying to hold a line instead of just 1. That might not work out so well for you.

I'm not sure what you mean. I assume you mean against 3 units of cavalry. If so then its just three versions of the 4 v 2 combats at the same odds and nothing changes. Did you mean something else? Again 3 attempts at (4 vs 2) is not the same as one at (12 vs 6).

While it is good to know the statistics, the larger question should always be, "How much damage will be caused should I fail?"

That's a bit glass half empty; "How often will I lose and what happens if I do?" are equally important questions. Or even better "How often will I win and what happens when I do?".

My FOG experience has been that the initial failure is quickly followed by further failure and so you're far better off fully understanding how often you're likely to win/draw/lose or you'll be unnecessarily timid.

Interesting blog!

Thanks John.

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Martin Stephenson
vexillia.blogspot.com
amazon.co.uk/shops/vexillia

If you look at three light horse units individually, we could, for argument's sake, have a 30% chance of victory for each indivisdual unit. No unit influences the next as a rule as probablility of one outcome is independant of another. However, if you look at it in terms of what are my chances that all three will win, then it becomes .3X.3X.3 or about a 2.7% chance of that happening…significantly less than the 30%.

However, you are correct in that you must know what the severity of the outcome is before you can truely assess it. I think that is why I said "How much damage will be caused should I fail?" in my original post!

And one more thing…

That's a bit glass half empty; "How often will I lose and what happens if I do?" are equally important questions. Or even better "How often will I win and what happens when I do?".

I am most concerned with the negative aspects of any outcome. There is little in the way of possitive aspects that would prevent me from performing an attack! Other options maybe but no possitive aspects!

John

If you look at three light horse units individually, we could, for argument's sake, have a 30% chance of victory for each individual unit. No unit influences the next as a rule as probability of one outcome is independent of another. However, if you look at it in terms of what are my chances that all three will win, then it becomes .3X.3X.3 or about a 2.7% chance of that happening…significantly less than the 30%.

I see what you mean now. I have two observations:

[1] The maths works against the cavalry too: see later.

[2] There's potential trap here: a 2.7% chance of the light horse losing all three does not mean there's a 97.3% chance of the cavalry always winning.

you are correct in that you must know what the severity of the outcome is before you can truly assess it.

So to applying this approach to your example. If the odds of the cavalry winning are 66% which include 33% winning by one and 33% by 2 or more (in rounded odds).

[1] The chance of the cavalry winning all three by one, or all three by 2 or more, is the same as the light horse winning: 0.33 x 0.33 x 0.33 = 0.036.

[2] The chance of the cavalry winning all three by any score is much better at 0.66 x 0.66 x 0.66 = 0.29.

So the great majority of outcomes lie in between and include partial wins for both sides. In FOG these only have a negative outcome if you also fail the cohesion test and any death rolls which further decrease their likelihood.

I am most concerned with the negative aspects of any outcome.

I can tell. :-)

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Martin Stephenson
vexillia.blogspot.com
amazon.co.uk/shops/vexillia