Thursday, 9 September 2010

Dice & Field of Glory

I posted a link to my previous probability article on the Yahoo! FOG list and as part of the resulting discussion Gino asked this “Is it better to have 6 dice needing 5s to hit re-rolling 1s or 6 dice needing 4s to hit?”.

I thought AnyDice would answer this quickly and so it proved. For a single dice the probabilities are straight forward and the roll needing 4s is the best option at 50% hits compared to 38.9% for 5s re-rolling 1.   Racking this up to 6 dice vs 6 dice produces these distributions:

So the average score for 5s re-rolling 1s is 2.33 compared to 3.00 for 4s without any rerolls.  The probabilities favour the straight roll by 1 hit.  This is confirmed if you run the two situations against one another as you would in a combat:

As expected the most likely outcome is a 1 hit loss for the battle group needing 5s to hit and re-rolling 1s.  The next most likely is  draw followed by a 2 hit loss.  To boil it down even further the probabilities are:

Outcome for battle group needing 4s to hit
Outcome Win Draw Lose
% 54.2 21.1 24.7

As I’ll explore in a future post, it’s worth noting that the probabilities of extreme results are very low because of the relatively high number of dice involved; in this case 6 a side.

If you want to experiment with this use this link to go straight to an AnyDice page. The code used was:

\Function code\
function: reroll N:n below M:n
{if N < M {result: [highest of d6 and N]} result: N }

output 6d [{5..6} contains [reroll d6 below 2]] named "A: 5s to hit re-rolling 1s"
output 6d [{4..6} contains [reroll d6 below 1]] named "B: 4s to hit no re-rolls"
output 6d [{5..6} contains [reroll d6 below 2]] - 6d [{4..6} contains [re-roll d6 below 1]] named "A - B"

Thanks once again to Jason Flick for creating AnyDice

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