In other words, the remaining possibilities are the same in both scenarios (AS/AH, AS/2C, AS/2D), but the probabilities attached to these possibilities are different. In scenario 2 the observation that one of the cards is the ace of spades does more than rule out possibilities, it also shifts the probability mass from the AS/AH possibility toward the AS/2C and AS/2D possibilities, in such a way that the probability attached to the AS/AH possibility goes ‘back’ to what it was when there were five possibilities.
It shows that drawing little boxes corresponding to each possibility and carefully crossing out those that were contradicted by the evidence is only a poor approximation of Bayes’ theorem.
Well, drawing boxes and crossing out would work here if you explicitly have boxes for “how does the card holder answer the questions” or in this case “how does the card holder answer the “if you have aces, pick one at random, is it a spade”?
Yup.
In other words, the remaining possibilities are the same in both scenarios (AS/AH, AS/2C, AS/2D), but the probabilities attached to these possibilities are different. In scenario 2 the observation that one of the cards is the ace of spades does more than rule out possibilities, it also shifts the probability mass from the AS/AH possibility toward the AS/2C and AS/2D possibilities, in such a way that the probability attached to the AS/AH possibility goes ‘back’ to what it was when there were five possibilities.
It shows that drawing little boxes corresponding to each possibility and carefully crossing out those that were contradicted by the evidence is only a poor approximation of Bayes’ theorem.
Well, drawing boxes and crossing out would work here if you explicitly have boxes for “how does the card holder answer the questions” or in this case “how does the card holder answer the “if you have aces, pick one at random, is it a spade”?
It just takes a bit more detail.
I made a diagram already. Note that we were using ‘prefer’ to refer to the ace you pick when you pick between the two if you have both.