A tagged fish, and the pond it probably came from

Question

In Pond A, 25\% of the fish carry a research tag. In Pond B, only 10\% are tagged. You walk to one of the two ponds chosen at random by a fair coin and catch a single fish. It has a tag. What is the probability you were fishing at Pond A?

Accepted Answer

Each pond is equally likely to be your spot. Imagine 100 trips, about 50 to each pond: - Pond A: 50 	imes 0.25 = 12.5 tagged catches. - Pond B: 50 	imes 0.10 = 5 tagged catches. Of the 12.5 + 5 = 17.5 tagged catches, 12.5 came from Pond A: P(	ext{A} \mid 	ext{tagged}) = \frac{12.5}{17.5} = \frac{5}{7} \approx 0.714. A tagged fish is more than twice as likely from Pond A, so the catch raises your belief from 50 percent to about 71 percent. Takeaway. Fractional counts are fine in the natural-frequency picture; only the ratio between the two sources matters for the final probability.

A tagged fish, and the pond it probably came from

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