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I was just logging in to search for help for some battlesuit config. math hammering, and this is exactly what i needed.

Nicely done. Well laid out and easy to follow. Do you have any excel models out there that people can use to model expected outcomes?

I took your guide and built an Excel spreadsheet to model expected kills. I could probably add some more dropdowns so that someone could select Eldar as the target, and have it automatically populate the save and toughness characteristics, but right now, I'm leaving those all as free form. I might also add the point efficiency calcs you mentioned. In any case, here's the spreadsheet, let me know if this seems to calculate correctly

http://www.kan.org/michael/mkp/files/expected_kill_calculator.xls

You can attach files like spreadsheets to articles... just upload them as if they were an image, and then attach them in using whatever name you uploaded it as. Eg; [[Image:SomeSpreadsheet]]. They will then be appended to the article in question

I'll do that once I get a little bit of feedback and make it a little more polished.

I didn't do that, kind of on purpose. I was looking at the article as a way for people to be able to start learning to understand probability. I think that the probability stuff is more useful used as estimates, during games, as a way of predicting outcomes and making decisions.

As such, I think that adding a spreadsheet to this article would actually detract from the value of the article, in that people would not learn how probability works, and instead would just use the spreadsheet.

I'm not against having a spreadsheet available as another article in the theory category, but I don't think it works well with the approach I was taking for this one.

Absolutely agree that probability and likely outcome assessment needs to be a skill that you can apply on-the-fly. After all, people aren't bringing laptops to tournaments and modeling every possible outcome before they move units.

However, there are a ton of people out there that just aren't good with math and a model is a good way to let them play around with a couple of scenarios and develop some rules of thumb they can use for those on-the-fly calculation.

I totally agree that a model is useful. I don't think it should go in the same article as the basics of probability is all I was saying.

Fair enough

What happens when the armour save is bypassed becuase of the AP value of the weapon?

Then it's modeled as a 100% failed save

You cant say "20 shots * 1/3 = 6 hits". Probability of rolling 5+ six times on 20 d6 is only 0.7.

I have just added the section on 'Standard Deviation and Variance' to this excellent article written by Redbeard. Unfortunately though, it has to be a bit 'mathsy' to properly explain how it works. If it doesn't read clearly enough, please let me know and I'll go back in and re-edit it.

Cheers!

Well, I love math... I'm going to have to find the time to read through this whole thing and learn what you guys are getting at here

Couple of bits that jumped out at me.

Expected Result:
This section could do with a bit more explanation. It really shows you how to calculate the mean. The mean and mode (most likely) result are not the same in this situation. On any given attack the more useful 'expected result' would be the one that you are most likely to actually achieve, i.e. the mode. If you can work out the mode you don't have to worry about all the rounding mentioned in the uses section later on.

In the example of 10 guardsmen shooting marines it shows the expected result as 5/9. That is the mean average you would get. Unfortunately it doesn't really help much to take the mean and round it to a whole number, as most people would intuitively round it to the nearest whole number, 1 because it is over 0.5. In fact the most likely result is 0 (with approx 56% chance of happening).

So if I use the example given, I want to know what is the most likely number of marines that will die.

To get the mode, use largest whole number that is less than or equal to (number of attacks +1)*probabilty. So in the example given you have (10 +1)*(1/18) = 0.61. The largest integer less than this is 0, so 0 is the most likely result. If the result from the maths is an Integer itself then you have 2 results that are equally likey, the value you worked out and the 1 below it. So 17 guardsmen would give (17+1)*(1/18) = 1, so both 0 and 1 are equally likely results.

Standard Deviation and Variance:
This section seems a bit misleading to me. The approximate 68%/95%/99% rule for 1,2 and 3 standard deviations applies to normal distributions, not necessarily other distributions like this one. There are also other issues to do with converting the integer numbers 40k works with to a continous normal distribution. If you just use the mean and SD as calculated and apply it to 1 volley of attacks you will find examples where the 68%/95% rule breaks down. There is another rule with some strange name that escapes me at the moment, that says that for any distribution; at a minimum 50% of all results are within ~1.41 SD, 75% are within 2 SD and 89% within 3 SD.

Puree, thanks for the comments. To help explain why what has been put in the article, it's probably best to deal with your points individually. Just to say though, I only wrote the parts of the article on 'Standard Deviation and Variance', although I will try to explain why Redbeard did what he did in the other parts. If Redbeard wants to add anything to what I say below, that's fine with me!

puree wrote:Couple of bits that jumped out at me.

Expected Result:
This section could do with a bit more explanation. It really shows you how to calculate the mean. The mean and mode (most likely) result are not the same in this situation. On any given attack the more useful 'expected result' would be the one that you are most likely to actually achieve, i.e. the mode. If you can work out the mode you don't have to worry about all the rounding mentioned in the uses section later on.

Yes, that's true. However, in order to calculate the mode, you would need to calculate the probabilities of killing 0 marines, 1 marine, 2 marines, 3 marines and so on, up to whatever squad size you were interested in and then see which one had the highest probability. Whilst this would give you the "result that happens the most often", it would take a long time to calculate and for the common uses of mathhammer, this is needlessly complicated. Expected result will suffice for pretty much anyone using mathhammer to inform their army list choices, since what the prospective general is interested in is "Is choice A better than choice B for my army?", or "How will this choice hold up against that type of enemy model". Expected value is simpler to use and thus better in this situation.

puree wrote:
In the example of 10 guardsmen shooting marines it shows the expected result as 5/9. That is the mean average you would get. Unfortunately it doesn't really help much to take the mean and round it to a whole number, as most people would intuitively round it to the nearest whole number, 1 because it is over 0.5. In fact the most likely result is 0 (with approx 56% chance of happening).

So if I use the example given, I want to know what is the most likely number of marines that will die.

To get the mode, use largest whole number that is less than or equal to (number of attacks +1)*probabilty. So in the example given you have (10 +1)*(1/18) = 0.61. The largest integer less than this is 0, so 0 is the most likely result. If the result from the maths is an Integer itself then you have 2 results that are equally likey, the value you worked out and the 1 below it. So 17 guardsmen would give (17+1)*(1/18) = 1, so both 0 and 1 are equally likely results.

Again, for the situation of a casual gamer applying mathematics to help him choose a particular troop type, or see how a troop choice will fare against an opposition unit, Expectation is a simpler statistic to calculate and thus more appropriate.

puree wrote:
Standard Deviation and Variance:
This section seems a bit misleading to me. The approximate 68%/95%/99% rule for 1,2 and 3 standard deviations applies to normal distributions, not necessarily other distributions like this one. There are also other issues to do with converting the integer numbers 40k works with to a continous normal distribution. If you just use the mean and SD as calculated and apply it to 1 volley of attacks you will find examples where the 68%/95% rule breaks down. There is another rule with some strange name that escapes me at the moment, that says that for any distribution; at a minimum 50% of all results are within ~1.41 SD, 75% are within 2 SD and 89% within 3 SD.

The 68%/95%/99% boundaries are for normal distributions, although as your sample size increases, this type of probability distribution (Binomial) will approach a normal distribution. Nothing you have said is wrong, but don't forget that the purpose of this article was to help the casual gamer to be able to apply Maths to help them make the best choices in the hobby. The reason that I have used this simplification is to make it accessible to anyone who would be reading it. Essentially, you are right and I agree with you, but a simplification is necessary given the context and applicability of the article.

You clearly have a much better understanding of statistics than the average person and you have seen many of the simplifications that Redbeard and I have made in compiling the article. Hopefully this post helps to explain some of the reasons why the choices that have been made have been made. I'd be interested to hear any further comments you have about this.

As a side note to this, personally, I don't like to use mathhammer with any degree of complexity past "This troop has a higher stat-line than that one, so I'll use it against that big nasty gribbly approaching me", but that's just my own choice. I never use statistics to analyse which troops I'll put in my army, but if others want to, this article should help them to do that!

Puree, thanks for the comments. To help explain why what has been put in the article, it's probably best to deal with your points individually. Just to say though, I only wrote the parts of the article on 'Standard Deviation and Variance', although I will try to explain why Redbeard did what he did in the other parts. If Redbeard wants to add anything to what I say below, that's fine with me!

puree wrote:Couple of bits that jumped out at me.

Expected Result:
This section could do with a bit more explanation. It really shows you how to calculate the mean. The mean and mode (most likely) result are not the same in this situation. On any given attack the more useful 'expected result' would be the one that you are most likely to actually achieve, i.e. the mode. If you can work out the mode you don't have to worry about all the rounding mentioned in the uses section later on.

Yes, that's true. However, in order to calculate the mode, you would need to calculate the probabilities of killing 0 marines, 1 marine, 2 marines, 3 marines and so on, up to whatever squad size you were interested in and then see which one had the highest probability. Whilst this would give you the "result that happens the most often", it would take a long time to calculate and for the common uses of mathhammer, this is needlessly complicated. Expected result will suffice for pretty much anyone using mathhammer to inform their army list choices, since what the prospective general is interested in is "Is choice A better than choice B for my army?", or "How will this choice hold up against that type of enemy model". Expected value is simpler to use and thus better in this situation.

puree wrote:
In the example of 10 guardsmen shooting marines it shows the expected result as 5/9. That is the mean average you would get. Unfortunately it doesn't really help much to take the mean and round it to a whole number, as most people would intuitively round it to the nearest whole number, 1 because it is over 0.5. In fact the most likely result is 0 (with approx 56% chance of happening).

So if I use the example given, I want to know what is the most likely number of marines that will die.

To get the mode, use largest whole number that is less than or equal to (number of attacks +1)*probabilty. So in the example given you have (10 +1)*(1/18) = 0.61. The largest integer less than this is 0, so 0 is the most likely result. If the result from the maths is an Integer itself then you have 2 results that are equally likey, the value you worked out and the 1 below it. So 17 guardsmen would give (17+1)*(1/18) = 1, so both 0 and 1 are equally likely results.

Again, for the situation of a casual gamer applying mathematics to help him choose a particular troop type, or see how a troop choice will fare against an opposition unit, Expectation is a simpler statistic to calculate and thus more appropriate.

You do not have to work out all the individual chances of killing 0, 1,2 etc marines. As I explained the modal average is largest whole number <= (n + 1) * p. I'm not sure how the mean is a simpler statistic to calculate, mean is n*p, mode is basically (n+1) * p, surely the addition of 1 to n is not that much harder to do. If you can calculate the mean on the fly then you can just as quickly calculate the the mode, and given that the mode is the real 'most likely' result it saves you all the rounding off that was discussed in the last section (round down hits and wounds and up for saves).

Not to say that mean is not a useful number whilst building lists, but at the point of doing something during a game turn the mode is more useful.

Let's face it, either way, your answer is going to be very similar! In a situation like this, there is unlikely to be a significant difference between Mean and Mode.

I must admit that I'm not familiar with your (n+1)*p formula for the mode. The mode is defined as being the result that happens the most which, for discrete random variables, means the value that is most likely to happen, i.e. has the highest probability - hence the need to calculate all those probabilities. Could you post a link to a page quoting your formula as I'm quite intrigued by it and would like to read up on it. You're not thinking of the mode for a continuous random variable are you?

No, that is the mode for a binomial distribution. You can test it if you want by calculating it as you say and see if the above calculation agrees (spreadsheets for the win), else search on google. Wikipedia provides a lot of the formula to do with Binomial distributions, other sites will also have them , though most only bother to mention mean and variance.

The only issue with using Mean during a game is, as I said earlier, most who do not have a good grasp of stats will intuitively round to the nearest number, hence they often 'expect' to do better than the mean is telling them as they assume a mean 0.7 = will probably kill 1 when the reality is they are most likley to kill 0, they then claim 'below average' rolls lost them the game (good humouredly or not), when they didn't have below 'average' rolls - they just used the wrong average.

Doh! Yes. Just spotted that one. Sorry for the misunderstanding on my part. It's late here, blah blah blah, rubbish excuses for not thinking straight

I'm not sure if I agree with you about how people will interpret the value of the mean, but I can see your point. Again though, considering the uses of mathhammer and it mainly being used to answer one of two questions (1. Is unit A or B better in my army? 2. Is my unit better used by pitching it against opposition unit A or B?), it would seem that actually the mean is the more useful statistic. If we consider question 2 and we are looking at, for example, "Should I pitch my unit of 10 guardsmen at the unit of tactical space marines or the unit of terminators?", then the mode will be 0 for both, however there will be a slight difference in the mean, since the tac marines will have a lower armour value (Yes, I'm choosing an obvious example to illustrate my point). By seeing the slight difference in the mean, the player would know that they had a marginally higher chance of killing a tac marine than a terminator, whereas the modes would be the same and thus the player would have gained no insight into their problem. In fact, for most questions that do not have an answer as obvious as the one I have proposed, you'd probably find that the mode was the same in both cases (due to the discrete nature of modes) and the means would be different, thus more useful. If your question is simply "I put this unit against that unit, how many should I kill?", then yes, mode is very useful, but for any meaningful comparison to be made for the player who wishes to make army choices based on calculations (the intention of the article), mean is the only sensible choice.

Of course if, as you suggest, someone is doing this all during a game, you might find their opponent complains at them pulling out a calculator to calculate either mode or mean!

Good article overall and important for truly competitive games (e.g. tournaments).

I feel, however, it should be like playing poker, if you can't rough out your odds in your head, tuff-nuts as they say. No calculator time outs.

GoFenris wrote:Good article overall and important for truly competitive games (e.g. tournaments).

I feel, however, it should be like playing poker, if you can't rough out your odds in your head, tuff-nuts as they say. No calculator time outs.

Essentially, I agree with you, both in GW games and in poker. I'm lucky enough to be able to work the odds out while playing, in many situations, but to me that takes away from the enjoyment and so I prefer to not use maths too much, instead relying on the fun in 40k and the psychology in poker.

This is a great thing for before battle but i would get mad at a person if he/she stood there thinking about which unit should attack which units. (especially if their playing Orks).

Very interesting stuff overall, cheers.

 GoFenris wrote:
Good article overall and important for truly competitive games (e.g. tournaments).

I feel, however, it should be like playing poker, if you can't rough out your odds in your head, tuff-nuts as they say. No calculator time outs.

I agree and to me it is a refresher to what you can calculate in your head.

Good article and thanks!