I can't for the life of me figure this problem out:
"There is a 0.01% of chance that your area might be hit by an earthquake. You heard that if an earth quake is coming, 90% of times that immediately prior to the earthquake, animals such as frogs will leave their shelters and appear on the street. You also know that, there is 1% of chance for the frogs to appear on the street when there is no earthquake. One day you see a very scary scene on the street: hundreds of thousands of frogs appear everywhere. What's the probability that your area is going to have an earthquake?"
Let e = earthquake, f = frogs.
From the description, I've gathered:
Facts: p(e) = .0001, p(f | e) = .9, p(f ^ -e) = .01.
Goal: find p(e | f)
From those facts (since the summation of probability distribution over a random variable should be “1”):
p(-e) = .9999, p(f | -e) = 0.1
I know p(e | f) = ((p(e)*p(f | e)) / p(f)). We have p(e) and p(f | e), so we just need to find p(f). We have p(f ^ -e) and p(f | -e), which should help us find p(f). However, I've tried writing out all permutations of Bayes' theorem, and I can't seem to figure out how to solve for p(f). I think I must be missing something simple.
Can someone please point out what I'm missing?