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?