P(A|C) = P(A|B,C)P(B|C) + P(A|not B,C)P(not B|C).
You are a witness of a night-time hit-and-run accident involving a taxi in Athens. All taxis in Athens are blue or green. You swear, under oath, that the taxi was blue. Extensive testing shows that under the dim lighting conditions, discrimination between blue and green is 75% reliable. Is it possible to calculate the most likely colour for the taxi? (Hint: distinguish between the proposition that the taxi blue and the proposition that it appears blue.) What now, given that 9 out of 10 Athenian taxis are green?