Math-hammer, binomial distribution vs. the law of averages

[rgr_maddog]

I like the odder aspects of this hobby. One of them being math. I use it to help with unit selection and determining their roll or function within an army, but am having trouble with estimating performance.

 

For example, if I have 15 Blood Claws charging into a 10 man tactical squad, how many unsaved wounds are they likely to cause?

 

For scientific certainty I need a 95% confidence level. Binomial distribution suggests that of the 60 attacks the Blood Claws will deliver, 95% of the time they will land 24 hits, 97% of the time those hits will cause 8 wounds, and of those 96% of the time I can count on 1 kill.

 

Law of averages would suggest, they hit half the time, 30, they wound half the time, 15, and my opponent will fail 1/3 of his saves, for 5 kills on average.

 

I understand that it is more correctly stated in binomial distribution that 96% of the time I will have 1 or more kills, but that's not real comforting. I was wondering how some of you address the disparity presented by these two methods of prediction?

[Teetengee]

Generally, it is just a result of there being more attacks then there are things to kill. Therefore it over inflates the mean kills because the law of averages principle is taking into account all of those cases with more than 10 unsaved wounds. However, going for any sort of significance level isn't going to provide a meaningful expected result. To get an accurate expected result, you need to use the following equation.
Chance of 0 unsaved wound*0 + chance of 1 unsaved wound*1 + chance of 2 unsaved wounds*2 + chance of 3 unsaved wounds*3 + ...+ chance of 9 unsaved wounds*9 + chance of 10 or more* unsaved wounds*10= actual average number of wounds the enemy loses.
*or more: the or more step will always be the one that saturates the enemy's wound count.
The math gets far more complex when you have multiwound models and a chance of instant death (unless all wounds caused would be instant death, then it is just as if every enemy as one wound) or if you are fighting a vehicle as explodes results and stacked weapon destroyed/immobilized results raise the average.

[tdemayo]

I find that the law of averages works just fine in giving a rough estimate of a unit's capacities in a given circumstance. I want no more than a pretty simple metric I can do in my head as I play.

[Kaedes Nex]

Since you can rarely calc a 100% result for an action, I just use law of averages and assume it could be 50/50 worse or better than that.

 

If you roll poor, too bad. Your play based on the law of averages was reasonable and you just got screwed. If you do better, which you have a 50% chance of doing, then great.

 

The game does not allow enough time, casually or in timed competition, to compute plays that have 95% or higher rates of success. Every play you make you have to remember you can fail spectacularly since that's the nature of a dice game. Even if you walk within 2" and charge you can still whiff every swing and get slaughtered by those Grots. And it's often much more disheartening to make a plan with a 95% chance of success and fail than to simply go for a 50/50. It turns what would have been "a couple bad rolls" into "crushing defeat".

[rgr_maddog]

The game does not allow enough time, casually or in timed competition, to compute plays that have 95% or higher rates of success. Every play you make you have to remember you can fail spectacularly since that's the nature of a dice game. Even if you walk within 2" and charge you can still whiff every swing and get slaughtered by those Grots. And it's often much more disheartening to make a plan with a 95% chance of success and fail than to simply go for a 50/50. It turns what would have been "a couple bad rolls" into "crushing defeat".

I completely agree, and in game I cannot accurately crunch new numbers while surrounded by continually shifting variables. I'm talking more about testing likely scenarios unit X will face before getting to the table so I have some idea of what to expect and decide where I'm likely to get the best results out of unit X. Or in the list making stage to help me figure out if I need to drop the points for a power fist to keep a unit able to threaten a tank or can I get buy with a melta bomb or two.

 

Teetengee, I'll run the same situation through your equation tomorrow and see wha she looks like.

[Teetengee]

Keep in mind, that if the game has any since of points effectiveness balance (I know, that is a different can of worms) and if it is not designed to be rock paper scissor, no action you take should have  95% success rate at equal points between target and targeter.

[rgr_maddog]

Aww, but the game is rock, paper, scissors-ish- My heavy support beats your troops, my fast attack beats your heavy support, my elites beat your fast attack... and so on. I like knowing which enemy units are paper and which are scissors to my rock.

 

And there should be 95% certainty of plenty of things. If I'm charging you with the 15 blood claws I should be 95% certain I will get 1 kill, and probably more. That's not a bad thing, if it was completely random we'd all be playing candy land.

 

For ease of computation I ran the math a little differently, finding the product of the different probabilities of success (1/2 [hits] * 1/2 [wounds] * 1/3 [failed saves] = 1/12) instead of running each in separate distributions as described in my initial post. This way seems pretty accurate- chance of 1 unsaved wound 99.5%, 2 unsaved wounds 96.5%, 3 at 88.6%, and so on. However, putting only these few numbers in you equation:

chance of 1 unsaved wound*1 + chance of 2 unsaved wounds*2 + chance of 3 unsaved wounds*3

would look like (0.995*1)+(0.965*2)+(0.886*3) which equals 5.6, already suggesting a higher kill rate than the law of averages. I completed running the equation and wound up with an end result of slightly over 17, which is not the average number wounds my enemies have been taking. Am I doing something wrong?

[Teetengee]

All but the last term in the equation are the chance of getting exactly that number, rather than the chance of getting that number or better.