That lake fishing example is terrible without any indication what "most" really means. If I take that to mean that 99% of all fish in one of the lakes are red, then seeing one red fish already makes me pretty confident which lake it is from. There is also this massive assumption in whole thing that investing time to study or decide actually increases positive outcomes. Might need to show that and then see if time spe…
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 anywhere in between. How many fish do you need to catch to be relatively confident you've covered the 51% possibility? 2-3? 9-10? I'm sure there's a mathematic solution, but my gut tells me I'd want more than 2-3. Maybe as many as 10 or 20, depending on how frequ…
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.