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The probability of passing morale tests
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Old 27 Feb 2005, 14:11   #1 (permalink)
Shas'Vre
 
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Default The probability of passing morale tests

Warning: The following is heavy on statistics (though easy to understand ones). If you have an aversion to math, believe in astrology, or simply like to play by gut feeling then you may want to stop reading now. I am completely open to criticism if I've made a math error but arguing something using the "but I don't roll that way" or "what about luck?" approaches are pointless: The numbers do not lie.

I've started putting together various spreadsheets to work out the probabilities of certain events and to weigh the cost/benefit of various options. This started as an exercise for my GT ork biker army because of its unique make-up (nothing but bikes and buggies): I wanted to see how much I fire I need to concentrate to force morale checks and the probability of opponents failing those checks. Since my army is made up almost exclusively of St 5, AP 5 weapons this is of some interest b/c I anticipate having lots of problems with certain opponents. Thus I want to know, for example, what it's going to take to break a unit of HB devastators....

I thought I'd share some of the info I'm putting together since it directly affects virtually any army out there, Tau included. I've also thrown in some other analyses (which were a PITA to calculate) that are Tau specific which will be included later (as I find the time to post them).

So....
Leadership
Leadership is taken on a 2d6 which generates a perfect bell-shaped probability curve: In other words, you are much more likely to get scores closest to the middle of the range (6-8) and scores approaching the extremes become more and more unlikely.
Here's the probability table:


# of results
Result Out of 36 Probability Cumulative Probability

2 1 2.78% 2.78%
3 2 5.56% 8.33%
4 3 8.33% 16.67%
5 4 11.11% 27.78%
6 5 13.89% 41.67%
7 6 16.67% 58.33%
8 5 13.89% 72.22%
9 4 11.11% 83.33%
10 3 8.33% 91.67%
11 2 5.56% 97.22%
12 1 2.78% 100.00%

What do the columns mean? The first is the die result of adding the 2d6 together (2-12). The 2nd is the total # of ways you can get that results (out of a possible of 36). The 3rd is the probability (expressed as a percentage) of getting that result. The last column is the cumulative probability: This is the probability of getting a particular result or lower. Obviously this last column is the one that's pertinent for LD checks since you need to roll equal to or under a particular LD value to pass a test normally.

It is of interest to note: Contrary to popular belief which is based upon common sense, the most common result of rolling 2d6 is 7, not 6. This is b/c there are 7 different ways to generate a 6 from two dice, compared to just 5 ways to generate both a 6 or an 8.

So how does this apply to Leadership checks? Looking at the table you can see that you have a 58.3% chance of passing a test if your LD = 7. OTOH, raising a unit's LD to 8 (e.g., by buying a 'ui upgrade) increases this to 72.2% which IMO is a very good buy. ^-^

In my next post I'll show you the effects of adding an ethereal to your army. Stay tuned.... O0


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Old 27 Feb 2005, 14:30   #2 (permalink)
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Shas'El
 
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Default Re: The probability of passing morale tests

I await the next post! I've always wanted to know these sort of statistics as well as how much difference the thereal makes ;D
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Old 27 Feb 2005, 15:22   #3 (permalink)
Kroot Shaper
 
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Default Re: The probability of passing morale tests

Quote:
Originally Posted by 42
I await the next post! I've always wanted to know these sort of statistics as well as how much difference the thereal makes ;D
What he said. And thx for doing all the hard work for us...I was going to do something like this myself, but I was too lazy
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Old 27 Feb 2005, 15:46   #4 (permalink)
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Default Re: The probability of passing morale tests

*gets out the popcorn*

Start the show! Start the show!

I like statistics for causing moral checks, as you mentioned you were getting at for your bikes. The same holds true for us carbine users and rail rifle users (and it also effects us witch hunter users and space marine users that enjoy leadership attacks such as Fear of the Darkness and Divine Prounouncment).

Good stuff so far - Can't wait to see the rest
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Old 27 Feb 2005, 16:37   #5 (permalink)
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Default Re: The probability of passing morale tests

I'm glad to see there are people who appreciate my warped sense of "fun." The ethereal stats will be finished tonight; I just need to format them.

As an interesting side note: My bikes perform VERY similar to stealth suits statistically speaking: The range is the same, the fire output, the St, even the cost per model. My analyses so far have convinced me that stealth heavy is the way to go with my Tau. More on that later.
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Old 27 Feb 2005, 17:14   #6 (permalink)
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Default Re: The probability of passing morale tests

Quote:
Originally Posted by Waldo_Pepper
My analyses so far have convinced me that stealth heavy is the way to go with my Tau.* More on that later.
I knew it all along!
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Old 27 Feb 2005, 17:18   #7 (permalink)
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Default Re: The probability of passing morale tests

well i do have to say that its very hard to take into account probabilities of wiping out tanks...i tried once but its too hard, i assume you didnt take that into account when deciding stealths were best! Anyway...we all wait with anticipation of your new findings!
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Old 27 Feb 2005, 17:25   #8 (permalink)
Shas'Vre
 
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Default Re: The probability of passing morale tests

I got to this sooner than I thought I would.

Passing Morale Checks with an Ethereal

Out Cumul Reroll Cumul Reroll
Result of 36 Prob Prob Prob Probability

2 1 2.78% 2.78% 2.70% 5.48%
3 2 5.56% 8.33% 7.64% 15.97%
4 3 8.33% 16.67% 13.89% 30.56%
5 4 11.11% 27.78% 20.06% 47.84%
6 5 13.89% 41.67% 24.31% 65.97%
7 6 16.67% 58.33% 24.31% 82.64%
8 5 13.89% 72.22% 20.06% 92.28%
9 4 11.11% 83.33% 13.89% 97.22%
10 3 8.33% 91.67% 7.64% 99.31%
11 2 5.56% 97.22% 2.70% 99.92%
12 1 2.78% 100.00% 0.00% 100.00%


So the first 4 columns are the same as the first. It's the next two that are important for an ethereal. Rerolled prob is the increased prob of getting a particular result if you get to reroll failures. The cumulative reroll probability is the important one: It shows the chances of actually passing a morale check with an ethereal in your list. As you can see he has a HUGE difference: LD 7 with a reroll gives you nearly the same performance as a regular LD 9 and LD 8 with the reroll out performs LD 10 (take that CSM and Necrons!). More importantly the boost occurs in that critical 6-8 LD range so that you get the biggest benefit to the reroll. It's a 20-24% improvement (24% for 7, 20% for 8 ) in the chance to pass a morale check.

Here's the real kicker: Once your units drop below 50% and are at -1 LD (and are at risk of breaking for good), the ethereal's bonus makes the biggest difference. At LD 6 & 7, you're getting a 24% improvement in performance. Bonding is nowhere near as cost effective in keeping your units on the board (I'll eventually get around to posting an analysis of bonding too but that's for another day).

Simply put, the ethereal gives an army an enormous boost to their probability of passing a morale check. In fact, the ethereal makes a bigger difference than a Shas'ui upgrade for your FW teams but combining the two makes the biggest difference.

Ok, so I will let people digest this for a little while. I have a lot more thoughts on what this all means and how you can apply it to your army but I'll have to wait until I have a bit more time.

<i>P.S. - The stealth suit analysis has to do with all around performance. I'm not suggesting they're the whole army.... just that they're a lot better buy than most people think - so much so that I'm contemplating going..... gulp....crisis free. That's a discussion for a later post and time.
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Old 27 Feb 2005, 17:29   #9 (permalink)
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Default Re: The probability of passing morale tests

absolutly excellent work! *applauds*
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Old 27 Feb 2005, 17:37   #10 (permalink)
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Default Re: The probability of passing morale tests

*aplaudes* statistics are our friends they make us know stuff ;D.
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