Whimsical said:
I think that the math answer is good if you are willing to do it. I think that the Take 10 answer is not a good idea since the guards are not benefiting from having each them getting a check which is one of the value of having multiple guards in one area.
So why does standing near another guard make someone better at listening?
The benefit to multiple guards is primarily that if an alarm is raised, there is enough manpower to deal with the problem, as well as the benefit of being able to scan a larger area simultaneously. If you want your guards to be able to hear a sneaky rogue nearby, you need to hire more skilled guards, or ward the place with Alarm spells.
I agree with Auraseer--if they're just regular guards who don't know something's fishy, they'll be taking 10. If theyre on alert, figure out what they need to roll to succeed and roll as many dice as you have guards. If you clear the target number, they notice.
usdmw said:
Well, it's probably more trouble than it's worth, but you could just use the multiplicative property of probabilities. For example, if a character has a 50% of success on a Spot check (Spot +5, DC 15) and he needed to make 5 spot checks, his chance of succeeding on ALL of them would be:
50% ^ 5, or .5 * .5 * .5 * .5 * .5
Which means, for those for whom probability math is arcane, that the probability of succeeding on at least one would be equal to the 1-(the chance of failing on all of them).
So if your chance of failure is 50%, for five checks the chance of failing on all of them would be 50%^5, or 1/32 (about 3%). So the chance of succeeding on at least one is 97%. If your chance of failure is 75%, the chance of succeeding on at least one of five tries is about 76%. If the chance of failure is 80%, the chance of getting one is about 67%, etc.
So, if you want to account statistically for multiple guards making multiple rolls, you just raise the probability of failure to the power of the number of guards, subtract from one, and that's your percent chance. Not likely to map well to a d20, but you could roll percentile dice.