Fun with Bayes' Rule: Birds and Hamsters
Suppose that you've got a truckload of animals in boxes, but you can't actually see the animals. If the animal is a hamster, there's an 80% chance the box is green. If it's a bird, there's only a 20% chance the box is green. You're given a green box. How likely is it that the animal inside is a hamster?
Though it's tempting to answer that a hamster is most likely, we need more information. We have to know what proportion of the animals are hamsters.
Suppose that only 10% are hamsters. Then given it's a hamster (10%) there's an 80% chance the box is green. So 10% x 80% = 8% of the time we'll have a hamster and a green box. However, we'll also have a green box in 20% of the cases where the animal is a bird (90% of the animals). So 20% x 90% = 18% of the time we'll have a bird and a green box.
Bayes Rule says that the probability it's a hamster when you're handed a green box is:
Most likely, there's a bird in the green box. Notice that choosing a green box increases the probability it's a hamster (which you would expect just 10% of the time if you had no information about the box), but it's not enough to make a hamster the most likely expectation.