3 ms·

I'd wager that's part of the puzzle - lacking information, you jump to the conclusion that 99% are single color in a given lake. But, "most" could be 51% or an

• hannasanarion 5y ago
  The confidence is pretty easy to calculate with Bayes rule. p(redLake|redFish)= p(redFish|redLake)*p(redLake)/p(redFish). If you're doing multiple experiments, then you can say that p(redLake) is your belief and update it by multiplying by p(redFish|redLake)/p(redFish) every time you see a red fish, and its inverse whenever you see a grey fish. If both of the lakes are 49-51, the update size is 0.51/0.50 = 1.02. Each red fish you see should increase your confidence in the red lake by 2% over whatever it was before, and vice versa. Assuming your original assumption is 50-50: After you see one red fish, you can be 51% confident in the red lake. If you've seen 5 more red fish than grey, you can be 55% confident. If you've seen 10 more red fish than grey, you can be 60% confident. If you've seen 30 more red fish than grey, you can be 90% confident. If the proportion of fish in each lake is more substantial, your bayesian update is larger and your confidence increases faster. If each lake is 90% one color, a single piece of evidence should give you 90% confidence.