How to figure out who made the mistake

Question

Two biologists at a certain company are responsible for analyzing water samples. Biologist 1 analyzes 45% of all water samples while biologist 2 analyzes 55%. Biologist one makes a mistake 3% of the time while biologist 2 makes a mistake 1% of the time. If a particular water sample was analyzed incorrectly, what is the probability that it was biologist 1 who analyzed it?

Answer 1

It might help to set out the information in the form of a tree:

                         +-(0.03)-: (A) Mistake     (0.0135)
 +-(0.45)-: Biologist 1 -|
 |                       +-(0.97)-: (B) No mistake  (0.4365)
-|
 |                       +-(0.01)-: (C) Mistake     (0.0055)
 +-(0.55)-: Biologist 2 -|
                         +-(0.99)-: (D) No mistake  (0.5445)

The probabilities given in the parentheses on the right correspond to all the possible outcomes without knowing which of the two biologists performed the water analysis or whether there was a mistake made.

If we know that a mistake has been made then notice that events (B) and (D) are ruled out. The probability that remains given this information, then, is then 0.0135 + 0.0055 (events A + C), of which only 0.0135 (event C) represents the probability that it was biologist 1 that made the mistake. The probability that biologist 1 made the mistake, then, is:

$$\frac{0.0135}{0.0135 + 0.0055}=0.71$$

See Bayes theorem for more on "reversing the condition" type problems.