## Friday, July 13, 2012

### Probability of Failing Charges (including Fleet!)

Sadly with the demise of my local club I haven't been able to get in a game of the new 40k, so I haven't had much to post. I will say that I love most of the changes in theory, except the god-awful wound allocation rules and the game balance fucking of the flier rules. With all the new rules and dice rolling I thought I could help out by calculating the probabilities of a very important new mechanic.

Random charges have got to be up there in the list of most controversial changes in the new edition. Personally I'm on the fence, but then again I only play shooty armies. It's pretty harsh for assaulting armies to fail a charge. The rules now are that units roll 2d6 to determine their charge range (after a 6" move). Saving your jump pack for the assault allows you to reroll the dice. Fleet allows you either reroll both dice or just reroll one of them.

There are two questions to get an understanding of here, be it through intuition or math hammer:
• What is the chance of making a charge for a given distance?
• When a unit with Fleet fails a charge, when should you choose to reroll only the lowest die?
Well I have all the answers for you!

First of all, this table tells you the chance of failing a charge, both for normal charges and those enhanced by Fleet or jump packs. The Fail Combos column is the number of dice rolls that fail the charge, out of the 36 possible combinations on two dice. Note that in order to achieve the percentages shown in the Fleet column you must know when to reroll the lower die and when to reroll both dice, which I explain below.

 Distance Fail Combos Normal Fleet Jump Pack 3" 1 3% 0% 0% 4" 3 8% 1% 1% 5" 6 17% 3% 3% 6" 10 28% 7% 8% 7" 15 42% 15% 17% 8" 21 58% 28% 34% 9" 26 72% 43% 52% 10" 30 83% 60% 69% 11" 33 92% 73% 84% 12" 35 97% 91% 95% Chance of Failing a Charge

So lets have a quick look at this. Only a 28% chance of failing a 6" charge, but up to 58% for an 8% charge. The probabilities shift quickly in the middle of the charge. Interestingly Fleet has a 28% chance of fail at 8" and 60% chance at 10", which suggests you are getting something like a 2" bonus on everyday charges. Lets make that a rule of thumb:

Fleet roughly gives you an extra 2" of charge distance

There is a tonne of information in the above chart and its going to take a while before all this stuff becomes intuitive. Maybe worth printing out a copy for the short term?

Now the second table I have shows some of the the different combinations of dice you can roll on failures, and the probability of failure if you just reroll the lowest die. You need to look at the chance of failure when rerolling one die, and compare it to the chance when rerolling both dice, in order to determine which course of action to take. The Worthwhile column says whether the odds on the single die are better than rerolling both. The Y/N value means that the odds are identical.

Luckily (because there is no mathematical reason for this) the entire table can be summarised into a single rule of thumb:

Reroll the lowest die when the highest die is 4 or higher;
unless it has no chance of success

To be honest I am amazed that so much work and such complexity can be summarised so easily. Amazed - but relieved! Hope this is helpful to you all. As always, I have tried to keep the Mathhammer to a minimum :) For those of you who care, here is the second table. Its an ugly one:

 Distance Low Die High Die Combinations Reroll Fail % Worthwhile? 5" 1 3 2 17% Y/N 1-2 3 4 33% N 1-2 2 3 50% N 6" 1 4 2 17% Y 1-2 3 4 33% N 1-2 2 3 50% N 1 1 1 67% N 7" 1 5 2 17% Y 1-2 4 4 33% Y 1-3 3 5 50% N 1-2 2 3 67% N 1 1 1 83% N 8" 1 6 2 17% Y 1-2 5 4 33% Y 1-3 4 6 50% Y 1-3 3 5 67% N 1-2 2 3 83% N 9" 1-2 6 4 33% Y 1-3 5 6 50% Y 1-4 4 7 67% Y 1-3 3 5 83% N 10" 1-3 6 6 50% Y 1-4 5 8 67% Y 1-4 4 7 83% Y/N 11" 1-4 6 8 66% Y 1-5 5 9 83% Y 12" 1-5 6 10 83% Y

1. How about adding terrain and move through terrain for a more detailed discussion.

2. Great stuff, and second eriochrome's suggestion!

3. nice posting.. thanks for sharing.

4. Interesting.

Anyway, I figured I'd let you know that the Table Top Generals podcast has added a feed to iTunes.

Use the advanced tab in iTunes, use the 'subscribe to a podcast' option and enter our feed site for best results:

http://ttgpodcast.podbean.com/feed

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