Probability the one true fish

Question

I have exams in Machine Learning coming up and I need help answering this question.

There are a million identical fish in a lake, one of which has swallowed the One True Ring. You must get it back! After months of effort, you catch another random fish and pass your metal detector over it, and the detector beeps! It is the best metal detector money can buy, and has a very low error rate: it fails to beep when near the ring only one in a billion times, and it beeps incorrectly only one in ten thousand times. What is the probability that, at long last, you’ve found your precious ring?

This is my answer I worked out:

Is this the right way to work out this type of question and is that somewhat the correct answer?

Answer 1

What you're looking for is P(A|B), which is the probability of having the proper fish when the detector beeps.

The probability of the detector blaring if you have the proper fish is P(B|A) = 9999/10000. We're not sure if the fish you have is the right one, though. All you know is that the detector beeps, and you have no way of knowing whether it's a true positive (P(B|A) or a false positive (P(B|not A).

The Bayes theorem allows you to flip between P(B|A) and P(A|B), ensuring that the other data isn't wasted. You don't require all of it to solve the problem.