Question

a town has two lakes, in lake one half of the fish are red fish, half is blue fish. lake two 1/4 is blue fish, 3/4 is red fish. If one fisher take one blue fish to home,what is probability that the fish is from lake one

Answer 1

P( red | lake 1) = 0.5, therefore P( blue | lake 1) = 0.5

P( blue | lake 2) = 0.25 and P( red | lake 2) = 0.75

Assuming that the fish could be picked equally likely form the 2 lakes, we have here:
P( lake 1) = P(lake 2) = 0.5

using law of addition of probability, we have here:

P( blue ) = P( blue | lake 1)P(lake 1) + P( blue | lake 2)P(lake 2)

P( blue ) = 0.5*0.5 + 0.25*0.5 = 0.375

Using bayes theorem, probability that a blue fish is obtained from lake 1, is computed here as:

P( lake 1 | blue ) = P( blue | lake 1)P(lake 1) / P( blue )

P( lake 1 | blue ) = 0.5*0.5 / 0.375 = 2/3

Therefore 2/3 is the required conditional probability here.