Ogrork the Mighty said:
Let's say there is a 5% chance per hour of an encounter. What is the chance over an 8 hour period that an encounter will occur? I realize that each check is independent of the others, but isn't there a way to figure out an overall % chance for the period?
I'm lousy at probability, but this is one of the few I do know. I'm unlikely to be first to respond, because I'll try to explain as I go.
First, understand that the probability of two events both occurring is (chance of one event) times (chance of second event). The chance of rolling 12 on 2d6 is 1 in 36, which is (1 in 6) times (1 in 6).
Second, understand that something
not occurring is also an event. The chance of
not rolling a 6 on 1d6 is 5 in 6.
Finally, understand that the probability for all outcomes, summed, has to equal exactly 1.
Given these three things, you can say that for each of 8 hours, the probability of not having an encounter is 19 in 20. So, for
all 8 hours, the probability of not having an encounter is (19 in 20) times (19 in 20) ... and so on, or (19/20)^8. That comes out to 0.6634 or so (66%). Given that, and since you know that the (chance of not having an encounter) plus (the chance of having an encounter) has to equal 1, you end up with a probability of having an encounter of 0.3366 (34%) or so. Or close enough to 1 in 3 not to sweat the difference.