Which test should I use? I have a set of probabilities that an event will occur (yes/no). E.g. [0.87, 0.56, 0.97], and I need to know "What is that probability that at least X of these events occurs?". I've been looking into Binomial Probability Density and related tests, but they all seem to assume that the probabilities for each event are the same.
Thanks for your help! I hope this isn't too rudimentary for this forum.
 A: Tools like the binomial assume equal probabilities because that greatly simplifys things, so you cannot simplify.  If you need an exact answer then you will need to look at every possibly way that your event (at least X events occure), calculate the probability of each possibility then sum those up.  There are tools that will generate every possible combination for you so that you can sum them up (the combn function in R is one) so this is doable.
If the set of probabilities is fixed, so you don't need to recompute every time and can just precompute the chances, and you can live with a good approximation then I would suggest simulation.  Randomly generate an outcome with the given set of probabilities, repeat this a bunch of times (10,000 or even 100,000) and then just calculate how often you had X or more events.  This would take only a few lines in R (and probably be similar in other tools as well).
There is a similar question here, see my example there (using R) for the more exact calculation.
