Following up on an earlier post, here is another example of how thinking about conditional probabilities in terms of causation rather than logical dependence can lead to cognitive dissonance. This example comes from 'Entropy Demystified,' by Arieh Ben-Naim.

Let A, B, and C be three hypotheses, such that P(B|A) > P(B) and P(C|B) > P(C). Suppose I tell you that P(C|A) < P(C). How does this strike you?

Knowledge of A makes B more likely, and knowledge of B makes C more likely, so shouldn't knowledge of A necessarily also make C more likely?

If you are thinking along these lines and are struggling to accept that the situation I describe is possible, then I suspect it is because, despite my warning, you can't help thinking about P(x|y) in terms of causation. The specific property of causality that seems to invade our heads when we contemplate problems like this is its transitivity: if A causes B and B causes C, then A necessarily is the cause of C.

Logical dependence, however, is not required to exhibit this transitivity property. Let A, B, and C be hypotheses about the outcome of rolling a six-sided die, such that for each hypothesis, the face showing is one of the four specified:

A: {1, 2, 3, 4}

B: {2, 3, 4, 5}

C: {3, 4, 5, 6}

P(A) = P(B) = P(C) = 4/6. If you are told that A is true, then the probability associated with B becomes 3/4, which is greater than 4/6. Similarly, P(C|B) is also greater than P(C). But P(C|A) = 2/4, which is less than P(C), as promised.

Here's another example of unexpected intransitivity for dice, operating on a different level, taken from Ian Stewart's entertaining "Cabinet of Mathematical Curiosities."

Mr. A proposes to his friend Mr. B that they play dice for money with his special dice. He has three of them, and they are each to pick one to play with against the other. Whoever roles the higher number on each throw wins the stake. To convince Mr. B that the dice are fair, Mr. A insists that he take first choice from the 3 dice - that way, if one of the dice is better than the of the others, Mr. B should have an fair chance of picking the better one.

The dice do not have the usual numbers marked on their faces, though. The red one has the numbers {3, 3, 4, 4, 8, 8}. The yellow one is marked with the numbers {1, 1, 5, 5, 9, 9}. Finally, the blue one has the numbers {2, 2, 6, 6, 7, 7}.

Which one, if any, should Mr. B choose?

Check the numbers for yourself. Whichever die Mr. B selects, Mr. A can select one of the others that gives a higher probability of winning.

Here's another example of unexpected intransitivity for dice, operating on a different level, taken from Ian Stewart's entertaining "Cabinet of Mathematical Curiosities."

Mr. A proposes to his friend Mr. B that they play dice for money with his special dice. He has three of them, and they are each to pick one to play with against the other. Whoever roles the higher number on each throw wins the stake. To convince Mr. B that the dice are fair, Mr. A insists that he take first choice from the 3 dice - that way, if one of the dice is better than the of the others, Mr. B should have an fair chance of picking the better one.

The dice do not have the usual numbers marked on their faces, though. The red one has the numbers {3, 3, 4, 4, 8, 8}. The yellow one is marked with the numbers {1, 1, 5, 5, 9, 9}. Finally, the blue one has the numbers {2, 2, 6, 6, 7, 7}.

Which one, if any, should Mr. B choose?

Check the numbers for yourself. Whichever die Mr. B selects, Mr. A can select one of the others that gives a higher probability of winning.

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