Originally posted by AThousandYoung
I can't find your mistake. Can anyone else? Or can they find my mistake in case this solution is correct?
EDIT - I think the mistake is that you assume knowledge you don't have; that is, you don't know that the Mad Scientist reached into the jars randomly and pulled out a random organ. This assumption of yours was necessary for this step which was ...[text shortened]... ]The possible outcomes of drawing at random (with probability weightings) are as follows:[/b]
I think you have to assume the game is fair (i.e. the scientist picks randomly), otherwise you have to make several additional assumptions which are tenuous at best. Even if he doesn't, but you don't know about his scheme, you can still answer the question truthfully and correctly.
However, I think my original solution was wrong. This is a case of conditional probabilty, so we use the formula P(A given B) = P(A and B)/P(B), where A = 2 brains in one jar, and B = a brain was chosen. The possible outcomes and weightings (normalized this time) are as follows:
1) 2 brains, empty
brain (P=1/16), brain (1/16), empty (1/8)
2) brain, brain
brain (1/8), brain (1/8)
3) brain + liver, empty
brain (1/16), liver (1/16), empty
4) brain, liver
brain (1/8), liver (1/8)
The probability that there are 2 brains in the jar and a brain was picked is simply 2*(1/16) = 1/8 from (1) above. The probability that a brain was picked is (1/16) + (1/16) + (1/8) + (1/8) + (1/16) + (1/8) = 9/16. Now using the formula for conditional probability, we get:
P(A given B) = P(A and B)/P(B) = (1/8)/(9/16) = 2/9
My original solution did not take into account the chance of the brain being from the 2 brain jar. My brain must have been in a jar at the time.