I have had a mathematical question in mind for years now, and have finally managed to work it out. As we deal in statistical probabilities in the games we enjoy, I thought I would lay it on you, as it could help you work out your best options in a shooting or assault opportunity in a game. It may even confirm your own ‘gut instincts’ about the killingness of your own troops.

I have worked out a mathematical formula to calculate the percentage likelihood of causing a certain number of casualties with a set number of shooters at a set percentage. An example of my requirement was; what is the percentage chance of causing 7 casualties with 12 shots if the percentage of causing one casualty is 42% with each shot? I wanted a formula that I could use in a spreadsheet or palm computer that would give me this without fuss, once I had entered in the three variables.

Trying to get my own head around this, I found I could more easily grasp this simpler version; what is the percentage chance of causing one casualty with two shots if the percentage chance of causing one casualty is 50%? This example is exactly the same as ‘When tossing a coin twice, how likely am I to get one heads and one tails?’ Lets also take a Heads to indicate a casualty.

There are four possible outcomes: Tails + Tails (or No Casualties)
Heads + Tails (or One Casualty)
Tails + Heads (or One Casualty)
Heads + Heads (or Two Casualties)
Each of these four outcomes has an equal chance of occurring. There are Two chances in Four of having one Heads and one Tails (or of One Casualty) so Two divided by Four is ½ or 50%. Therefore, there is a 50% percentage chance of causing One Casualty with Two Shooters (each shooting with a 50% chance of causing one Casualty).

Okay. I am using an MS-Excel spreadsheet for this next part. By the way, we are dealing with the concept of Expected Values in the area of Statistical Probabilities, involving Binomial Distributions (I genuinely haven’t put this in to try to sound like a smartarse, I would be delighted if someone would check my workings. Maths is certainly not second nature to me, but I am prepared to have a stab)

We are using three variables: Shots (a whole number greater than zero)
CasReq (a whole number, this ranges from zero to the number of shots)
KillPC (between 0% and 100%)
and I am using named cell references in my spreadsheet formula.

The formula consists of three ‘chunks’ or factors: A coefficient (drawn from Pascals Triangle)
A factor drawn from the chance of the shot succeeding
A factor drawn from the chance of the shot NOT succeeding
The product (multiplication) of these three factors gives us the result

The coefficient is basically the number of rows down on Pascals Triangle as the number of Shots, and the number of ‘columns’ across according to the Required Casualties.

This is a link to a webpage on Pascals Triangle.

Pascals Triangle (http://www.firechicken.net/ptri/)

So in Excel, you could either create a Pascals Triangle on a separate sheet and index it like this:
=INDEX(Pascal,Shots+1,CasReq+1) (Pascal is a named range, the Shots+1 is because I have a 0 row, and CasReq+1 because we have to include the possibility of zero casualties occurring)

OR

Use this formula which involves the factorial function:
=(FACT(Shots)/(FACT(CasReq)*FACT(Shots-CasReq)))

The Success Factor is:
=(KillPC^CasReq)

And the Failure Factor is:
=((1-KillPC)^(Shots-CasReq))

Stick’em altogether in one formula:
=(FACT(Shots)/(FACT(CasReq)*FACT(Shots-CasReq)))*(KillPC^CasReq)*((1-KillPC)^(Shots-CasReq))

so, to go back to my sample question “What is the percentage chance of causing 7 casualties with 12 shots if the percentage of causing one casualty is 42% with each shot?�, entering

Shots = 12
CasReq = 7
KillPC = 42%

The result is 11.98 %

It is interesting doing the entire range of possibilities, to get a good idea of how killy your guys are gonna be (or not). This is the range of possible outcomes, against their likelihood for 12 shots at 42% kill chance (which happens to be the possibility of killing an Imperial Guardsman with a Tau Pulse Rifle) Casualties _ Probability
0 __________ 0.14%
1 __________ 1.26%
2 __________ 5.02%
3 __________ 12.11%
4 __________ 19.73%
5 __________ 22.85%
6 __________ 19.31%
7 __________ 11.98%
8 __________ 5.42%
9 __________ 1.75%
10 _________ 0.38%
11 _________ 0.05%
12 _________ 0.00%
(total) (100%)
Basically you are most likely to score 4, 5 or 6 casualties, in fact there is a 62% or nearly two chances in three that you will kill 4 or 5 or 6.

Another twelve shots, but at 11% (kill chance per Pulse Rifle vs. Marine): Casualties _ Probability
0 __________ 24.70%
1 __________ 36.63%
2 __________ 24.90%
3 __________ 10.26%
4 __________ 2.85%
5 __________ 0.56%
6 __________ 0.08%
7 __________ 0.01%
8 __________ 0.00%
9 __________ 0.00%
10 _________ 0.00%
11 _________ 0.00%
12 _________ 0.00%
(total) (100%)
The zeros lower down are just because there are only two decimal places showing. In fact there is roughly a 1 in 1,000,000,000,000 (i.e. one trillion) chance of obtaining 12 casualties.

So, there’s about a one quarter chance of no casualties, and less than a one in ten chance of killing four or above. So if a ten man Marine squad is stamping towards you from just outside the maximum range of your Pulse Rifles, and you are blazing at it with 12 rifles, you will have three shooting rounds before his own rapid fire Bolters are capable of returning fire. You might expect to not cause any casualties in one of those rounds, and you are half again as likely to cause one casualty as two. So assume about seven Marines are gonna survive until they get into close range. Better not move that Fireknife Crisis team too far away then...

Ok i sorta get that and i sorta dont... ill reread it when i have more time and expand on my opinion of this.

Ya man, I like my good ol' fractions.


And I'm suddenly overwhelmed with a feeling of hate for this "pascal" chap...

i wish we would take probability in math 20, but teacher says we are skipping that section entirely

I d0n't think y0u sh0uld apply pr0bability mechanics t0 40k at all, it's just n0t true, n0 matter h0w thinly y0u slice the numbers d0wn. Like it 0r n0t, three Fire Warri0rs will n0t always stand up t0 2 SMurfs, n0 matter h0w g00d the chances are in their fav0r. The same applies t0 all aspects 0f the game, and the m0re y0u try t0 cut and dry it, the w0rse y0ur luck ingame will be.


Just th0ught I'd add my seven 0r s0 sense.

Originally posted by Pandora@Apr 7 2004, 22:43
I d0n't think y0u sh0uld apply pr0bability mechanics t0 40k at all, it's just n0t true, n0 matter h0w thinly y0u slice the numbers d0wn. Like it 0r n0t, three Fire Warri0rs will n0t always stand up t0 2 SMurfs, n0 matter h0w g00d the chances are in their fav0r. The same applies t0 all aspects 0f the game, and the m0re y0u try t0 cut and dry it, the w0rse y0ur luck ingame will be.


Just th0ught I'd add my seven 0r s0 sense.
See here (http://forum.librarium-online.com/index.php?showtopic=8484)

I thought it was interesting, and could be really handy in a game situation if you had all the things you needed listed down. But then you'd probably spend more time flicking through your notes than playing the game. Besides, half of the fun is in the surprise ;)

Also, im in agreement with Oplopanax, Pandora, why dont you change your name to pand0ra?

Bloody hell. well done for working that out. But I don't understand the 0.14% chance with no shots...

oh thanks..you just shorted out my brain..ugh..math <_<

So, are you taking into account the "save throw" or just that a shot hits and wounds? I'm in Brief Calc, which deals a lot with MS Excel and its uses. I could probably write something up about it.


I d0n't think y0u sh0uld apply pr0bability mechanics t0 40k at all, it's just n0t true, n0 matter h0w thinly y0u slice the numbers d0wn. Like it 0r n0t, three Fire Warri0rs will n0t always stand up t0 2 SMurfs, n0 matter h0w g00d the chances are in their fav0r. The same applies t0 all aspects 0f the game, and the m0re y0u try t0 cut and dry it, the w0rse y0ur luck ingame will be.



Right, we all know that the game is more than just "numbers" and probablity, but we are taking a straight look at the whole number crunching aspect of it. If we really wanted to "simulate" a "real world" attitude of warfare, we wouldn't have point values attached to each unit to keep the teams "fair" now would we? I certainly don't think the Tyranids would care if the Space Marines have an equal number of points to make the battle fair ;).

No matter how much you people apply math too it, when it comes down to it, it is a matter of the dice. If you roll bad less die, if you roll good, more die.

No matter how much you people apply math too it, when it comes down to it, it is a matter of the dice. If you roll bad less die, if you roll good, more die.

Haha, but that's our point! :D. Aren't the end results of dice just numbers? Aren't there 6 sides to a dice, and won't a resulting die roll of being over or under a certain number yield a particular outcome? That's the basis of probability/statistics :D. We aren't "applying math to it" b/c math is already applied to it. We are just taking the game at it's most simple, basic of forms and giving you an idea of what you're looking at.

The game is formed around numbers, and then coupled with tactics, game environment (cover), and of course, luck. :D

But make no mistake, this is a mathematically based game. You can have the numbers, true, and know them inside and out, but if you don't have tactics, you're screwed. But we're just looking at the statistical aspect of it ;).

Wow. seriously, wow. you should right a book for GW.