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View Full Version : Math-Hammer : Statistical Dice Analysis

Aaron Chapman
28-07-2009, 04:23
I've compiled a full list of critical dice roll statistics for Warhammer Fantasy.

You can download it here : http://www.RemanLegions.com/Math-Hammer.pdf <--updated with new material 7/29/09

This document is designed to be a simple quick reference chart for use during game-play, so it will be lacking a few complicated situations such as combinations of multiple attacks and CR to determine average chances of success. Calculations like that require more than a simple chart.

The list covers:

* % chance to pass Leadership tests with "LD 2" - "LD 10" (including BSB re-roll & Cold-Blooded & Rapturous banner)

Magic-
* % chance to successfully cast or dispel a spell with Power Level 3-15 on 1D6-5D6
* % chance to cast a spell with Irresistable Force (or Auto-Dispel) on 1D6-5D6 (including Book of Hoeth & Skaven "13")
* % chance to Miscast on a spell (or Auto-Fail Dispel) on 1D6-5D6 (including Ring of Hotek)
* The suggested # of dice to roll for each power level

Shooting, Combat, Hits, wounds, Saves, etc-
* % chance to successfully rolling 1D6 needing 2+ - 9+ (including re-rolls)
* # of dice required to get an average of 1 successful D6 roll (including re-rolls)

Flee/Pursuit-
* % Chance to escape any combination of infantry/cavalry/dwarfs/skaven when fleeing from combat

************

If you have any suggestions for stats to be added please PM me and I'll see if I can add it in.

Unuhexium
28-07-2009, 07:14

Avian
28-07-2009, 08:55
Far too slow, I have had most of those things on my website (http://www.avianon.net/subpage.php?s=index_calculations) for years. And you are notably lacking something like this: Escaping Pursuit (http://www.avianon.net/calculations/pursuit.php). ;)

* # of dice rolls needed to guarantee 1 hit with shooting needing 2+ - 9+ to hit (including re-rolls)
* # of dice rolls needed to guarantee 1 wound in combat needing 2+ - 6+ to hit (including re-rolls)
* # of dice rolls needed to guarantee 1 wound with shooting & combat needing 2+ - 6+ to wound (including re-rolls)
Ain't no such thing.

mdiscala
28-07-2009, 13:27
I was in Advanced Calculus in highschool... now I cant do long division. I'll take a look at it though maybe its not as hard as it sounds.

28-07-2009, 15:37
Hey i like it, great work. I was looking for a similar page on the net last month. Couldn't find any decent ones. I've saved that as a favourite page :)

I assume your 9+ to hit is worked out needing a 6, then another 6

So 4d6 for casting the waaagh spell (11+ to cast, with staff baduum) has 13.1% of a miscast, and a 79.4% of casting (12.7% chance of IF) I'll take those odds

Troah
28-07-2009, 22:19
Ah, so this is Math hammer.

Aaron Chapman
29-07-2009, 01:56

Check out the updated version!

dapredator66
29-07-2009, 02:17
Far too slow, I have had most of those things on my website (http://www.avianon.net/subpage.php?s=index_calculations) for years. And you are notably lacking something like this: Escaping Pursuit (http://www.avianon.net/calculations/pursuit.php). ;)

Ain't no such thing.

I am sure you haven't got my Excel sheet for calculating spell casting probabilities for WoC. It can takes into account Tendrils of Tzeentch, Blood of Tzeentch and Conjoined Humonculus. Not really easy to calculate otherwise if your using 5 dice!!!

Artemis360
29-07-2009, 03:37
Dude, is there any chance you could be kind enough to add an extra calculation next to ring of hotek and book of hoeth about tendrils of tzeentch (re-roll one power dice/dispel dice per player turn). I'd really like to know the miscast/irresistable probability on that but I'm not smart enough to figure it out myself :(

Urgat
29-07-2009, 03:53
Math hammer is cool, heh? Too bad dies don't care about it, since their odds of rolling whatever are reset after each roll :p I don't think I've ever had a game where my rolls followed those supposed "averages".

AvariceX
29-07-2009, 05:02
I've been thinking of doing the exact same thing for some time so hats off to you for actually doing all this work and letting me continue to sit on my lazy ***.

Seriously though, great work :D

Trains_Get_Robbed
29-07-2009, 07:06
You are a great man that has done what I and others consider tedious and boring. This won't go unappetciated! :)

Avian
29-07-2009, 08:18
Math hammer is cool, heh? Too bad dies don't care about it, since their odds of rolling whatever are reset after each roll :p I don't think I've ever had a game where my rolls followed those supposed "averages".
Math doesn't tell you what is going to happen, it tells you what the odds are of it happening. If you believe that the average result is what you are going to get, you have seriously misunderstood the subject.

I am sure you haven't got my Excel sheet for calculating spell casting probabilities for WoC. It can takes into account Tendrils of Tzeentch, Blood of Tzeentch and Conjoined Humonculus. Not really easy to calculate otherwise if your using 5 dice!!!
It is in fact impossible to give a definitive answer in some cases since these depend on player choice. A risk taker might use a Conjoined Homunculus when he got a score of 2 or even 3 below what he needed, while a more careful player might only use it when he was 1 below, for example.

Dragune
29-07-2009, 09:13
Thanks for posting this. I've always believed that sometimes you just have to play the percentages and have to accept the results. Good info to have :)

Urgat
29-07-2009, 11:35
Math doesn't tell you what is going to happen, it tells you what the odds are of it happening. If you believe that the average result is what you are going to get, you have seriously misunderstood the subject.

I'm pretty su're you're misunderstanding my post on purpose, so I won't insult you by explaining, m'kay?
edit: Not wanting to devaluate the OP's work btw, I'm just stating the truth of my games. If I roll say two hundred times during a game, I should expect around 1/6 of 6, and I get many games w/o me rolling a single 6. That's the truth behind math hammer: you can't make averages out of something random, when there's so many other parameters to consider (how the die is held, how it is thrown, on what it lands, if the die is one of those super imbalanced GW dies, etc etc). The dies just don't follow any mathematical rules but those of physics, and those can't be controled by our meager means (her... 6 faces so 1 out of 6!). Well, I ended up explaining anyway, but I didn't want to come across as rude, and when I read my previous post again, I did sound rude, so apologies.

Aaron Chapman
29-07-2009, 13:22
I wasn't planning on doing specific lores and army books, just the more common occurances in warhammer for quick reference.

Book of Hoeth & Ring of Hotek have multiple instances where the same effect will be seen from other items and they are so common and effect all wizards that it was reasonable to expect these stats were needed.

But I'll look into the odds of Tendrils & Homunculus unless someone has already figured them out. But as Avian has said there are several variables based upon how you use it to take into account.

dapredator66
29-07-2009, 14:57
It is in fact impossible to give a definitive answer in some cases since these depend on player choice. A risk taker might use a Conjoined Homunculus when he got a score of 2 or even 3 below what he needed, while a more careful player might only use it when he was 1 below, for example.

Of course you are right. However, it is nice to have an idea of the success rate if you would utilize these gifts/magic item whenever you needed it.

For example did you know that Blood of Tzeentch is very effective when you only throw 2 or 3 dice but that it is worthless if you throw more.

Avian
29-07-2009, 15:15
Having crunched the numbers and believing I have them right, I get the following for casting Gateway with Tendrils, needing to roll a 14+ on 5 dice:
Chance of getting it off: 86.8% (75.2% normally)
Chance of miscasting: 10.0% (19.6% normally)

I had a look at dapredator66's Excel file and he got slightly different numbers than I got, but I have no clue how macros work and thus I haven't been able to look at his math.

EDIT: Figures given were for rolling 15+, which is a bit irrelevant since you will be getting at least a +1 to cast. Have corrected them to show the figures for rolling 14+.

Aaron Chapman
29-07-2009, 15:30
Having crunched the numbers and believing I have them right, I get the following for casting Gateway with Tendrils, needing to roll a 14+ on 5 dice:
Chance of getting it off: 86.6% (71.0% normally)
Chance of miscasting: 10.0% (19.6% normally)

I had a look at dapredator66's Excel file and he got slightly different numbers than I got, but I have no clue how macros work and thus I haven't been able to look at his math.

A quick 5 minute look gave me roughly the same estimation.

ZeroTwentythree
29-07-2009, 15:54
If I roll say two hundred times during a game, I should expect around 1/6 of 6, and I get many games w/o me rolling a single 6. That's the truth behind math hammer: you can't make averages out of something random, when there's so many other parameters to consider (how the die is held, how it is thrown, on what it lands, if the die is one of those super imbalanced GW dies, etc etc). The dies just don't follow any mathematical rules but those of physics, and those can't be controled by our meager means

It's not about the odds of a single die rolling a particular result. That is always 1 in 6, as you say.

There are several ways in which statistics helps predict likelihood of success/failure.

In the case of a single die roll, we're not looking for a single result, but a result either above or below a particular number. So if you need a 5+ to wound something, that is a (roughly) 33% chance of success, and a 66% chance of failure. That is not the same as "random" and it does follow mathematical rules. Otherwise there would be no difference between needing a 3+ or a 5+ to wound.

When you need to roll multiple dice to wound, you are just taking those odds and grouping them. So with the 5+ there's a 33% chance of success on a single die, and 66% chance of failure. If you are rolling 2 dice, you have a limited number of results on the dice, a certain number of those will be no successes, a certain number will give you 1 success, and a certain number will give 2 successes. The number of each of these results can be expressed as a percentage (or fraction, or however you prefer to consider your odds.) Same with any other number of dice.

So again, each individual die still has a 1/6 chance of rolling any of the 6 faces, but there are specific odds on whether or not those numbers will result in success or failure for you.

Predicting how likely you will be in rolling above & below a target number on multiple dice that are added together (as when casting spells) works in a similar manner.

A simple graphical exercise to help visualize this would be to list all of the combination possible with a given number of dice and see how many result in success or failures. If you're bored and have some time. ;)

That is actually the truth about math hammer. ;)

Kerill
29-07-2009, 15:58
I am sure you haven't got my Excel sheet for calculating spell casting probabilities for WoC. It can takes into account Tendrils of Tzeentch, Blood of Tzeentch and Conjoined Humonculus. Not really easy to calculate otherwise if your using 5 dice!!!

Nice, but I think there are some errors in it. I noticed for 4 dice at a 15+ spell there are equal chance of IF and miscast, but in fact it's possible to roll for example 1,1,6,6, which would be a miscast- the miscast potential should be higher than the IF at this number of dice.

dapredator66
29-07-2009, 16:23
Having crunched the numbers and believing I have them right, I get the following for casting Gateway with Tendrils, needing to roll a 14+ on 5 dice:
Chance of getting it off: 86.6% (71.0% normally)
Chance of miscasting: 10.0% (19.6% normally)

I had a look at dapredator66's Excel file and he got slightly different numbers than I got, but I have no clue how macros work and thus I haven't been able to look at his math.

You have probably the most accurate answers. The macros are just throwing virtually the dice for you and then analyzing the results. It currently does 100 000 dice throws. This gives results accurate to about 1%.

If anyone is interested this method is called Monte Carlo simulation (http://en.wikipedia.org/wiki/Monte_Carlo_method).

Nice, but I think there are some errors in it. I noticed for 4 dice at a 15+ spell there are equal chance of IF and miscast, but in fact it's possible to roll for example 1,1,6,6, which would be a miscast- the miscast potential should be higher than the IF at this number of dice.

Yep, there should be a difference but the program is not accurate enough to show it (1%).

Avian
29-07-2009, 16:52
Ah, I see, that explains it.
Originally I actually wrote a simulation script in PHP as well, but when I didn't get accurate results I did a very rough algorithm that went through all the possible results and checked them.

http://www.avianon.net/calculations/tendrils2.php

This shows the number of possible combinations for each value when the threshold is 15. It's not polished, so you'd have to do the percentages manually and you can't alter the threshold without editing the script.

For example did you know that Blood of Tzeentch is very effective when you only throw 2 or 3 dice but that it is worthless if you throw more.
Well, to be more specific, it is most effective when the level you need to reach is approximately the average you'd roll on the dice. So with two dice, it is most effective for spell difficulties of 6-8. The reason it's not very effective for more than 3 dice is that there are few spells that have a casting value as high as the average on 4 or 5 dice.

And what it also shows is that it's not a very effective item. Look at your graph and you see that the curve for one dice more without the Blood of Tzeentch is essentially always above the curve for the same number of dice with the Blood of Tzeentch. Given that you can only use either once per turn and the Power Familiar is 5 pts cheaper, there is no reason to take the Blood.

People probably wouldn't be able to see that without the math. ;)