What happens if, for example, you know that this month is October and historically days in October have a 10% chance of rain, but you also know that the clouds were just sprayed with some rain-inducing chemical that historically gives a 70% chance of rain. Furthermore, the weatherman predicts rain today and he is correct 60% of the time. Is there a way to combine all this knowledge to get the probability that it will rain today?

Initially, the probability that it will rain is 0.1
With the chemical, this becomes 0.7.

Now, there is an independent weatherman, his prediction is 60% correct and he predicts that it will rain.

P(it rains AND weather man correct) = P(it rains) x P(weatherman correct)

I think you can work this out easily :)

If all three things are completely independent from each other, then this is pretty easy. You'd just probability that it WOULDN'T rain, then subtract that from 100%. It wouldn't rain if it just happened to be one of the 90% of days in October when it doesn't rain AND it was one of the 30% of times that the cloud-seeding chemical didn't work AND it was one of the 40% of times that the weatherman is wrong. That works out to 0.9 * 0.3 * 0.4 = 0.108 or a 10.8% chance that it won't rain. That means there's an 89.2% chance that it WILL rain.

Things get much more complicated (and impossible to answer without more information) if the events you describe are NOT independent. For example, perhaps the weatherman is aware of the rain-inducing chemical application, and that has influenced his decision. If so, how much of a factor was it in his prediction? Or maybe the chemical causes rain 70% of the time, but it's only applied on days when the relative humidity is above 80% or else it won't work. Presumably the chances of rain in October are much higher than 10% if you narrow it down to days with 80% or higher relative humidity.

Anyway, the list is practically endless of ways in which these events/observations could correlate with each other (i.e. not be independent), so it's a difficult question to answer in reality. But if your question is simply about finding the overall probability of several independent events occurring, use the technique I showed above.

Sorry, Jerry. Didn't see your answer until after I answered. Stupid GO skin! I like to use it to see all of the new questions in the Science topic simultaneously, but as you know it has a few issues. :)

Yea, it's okay though :)

jcaron2,
Thanks for the answer. I'm really interested in non-independent events where there is some cross-correlation. Maybe it would only naturally rain 5% of the time in October, but they spray the clouds more during that month so it's 10%. So, my real question is, supposing I was able to characterize those dependencies somehow, how would I calculate the probability that it rains?

If you can quantify the correlation, most inter-related events can usually be statistically modeled using some combination of conditional probability and marginal probability (along with a few tools, such as Bayes Theorem), all of which you can look up on Wikipedia.