While reading Think Bayes by Allen Downey, I encountered an exercise problem bellow.
Exercise: M&M's are small candy-coated chocolates that come in a variety of colors. Mars, Inc., which makes M&M's, changes the mixture of colors from time to time. In 1995, they introduced blue M&M's.
In 1994, the color mix in a bag of plain M&M's was 30% Brown, 20% Yellow, 20% Red, 10% Green, 10% Orange, 10% Tan.
In 1996, it was 24% Blue , 20% Green, 16% Orange, 14% Yellow, 13% Red, 13% Brown.
Suppose a friend of mine has two bags of M&M's, and he tells me that one is from 1994 and one from 1996. He won't tell me which is which, but he gives me one M&M from each bag. One is yellow and one is green. What is the probability that the yellow one came from the 1994 bag?
I though that there were two hypotheses: one was that the yellow M&M was from a bag from 1994 and the other was that the yellow M&M was from a bag from 1996. Although I was not really sure about an evidence, I guessed that the evidence is that one M&M is yellow and the other is green.
However, according to the solution, those hypotheses are defined like below.
Hypotheses
A: yellow from 94, green from 96
B: yellow from 96, green from 94
Then, prior probability for each hypothesis is 1/2. I also do not understand why P(yellow from 94 and green from 96) is 1/2. Should it be 0.2*0.2 = 0.04?
I think that I miss something very significant in my understanding.
My questions:
- Why do I need to include the green M&M information in the hypotheses?
- Why is the prior probability for each hypothesis 1/2?
- What would be the correct evidence I should use in this problem? In my understanding, hypotheses must not include any additional information. But, hypotheses in the solution certainly include additional information that one M&M is yellow and the other is green.