Probability 101 in New Logic/Math Puzzles Posted July 2 · Report reply I agree completely with EventHorizon, and will try to summarize in a way that addresses the OP and deconvolutes the paradox at its heart: Spoiler If Plato were to pick a number, say 4, and decide that he would tell you "You rolled a 4" or "You didn't roll a 4", then Aristotle would actually be correct. There's an 11/36 chance that he would roll a 4, and among those outcomes there are 2 that sum to 7, so if Plato says "you rolled a 4" then there really is a 2/11 chance that the sum is 7. Alternatively, there's a 25/36 chance that he doesn't roll any 4s, and among those outcomes there are 4 that sum to 7, so if Plato says "you didn't roll a 4" then there's a 4/25 chance that the sum is 7. So Aristotle is slightly more likely to have rolled a 7 if Plato says he rolled a 4 (or any other number). The reason for this is that if Aristotle rolls doubles then it obviously won't sum to 7, and Plato is slightly less likely to say that Aristotle rolled any given number if he rolled doubles (1/6 chance of rolling the number Plato picked) than if he rolled two different numbers (1/3 chance of rolling the number Plato picked). On the other hand, if Plato picks one of the two numbers that are rolled and calls it out, the math is different. If Aristotle rolls two different numbers then there's a 1/2 chance that Plato will call out either of the two numbers, but if he rolls doubles then there's a 1/1 chance that Plato will call out that number. So there are 11 possible rolls that could lead to Plato calling out 4, but one of those possibilities (rolling double 4s) is twice as likely to lead Plato saying that he rolled a 4 as the others, so the odds of rolling a sum of 7 should be 2/12 instead of 2/11. This fits with our intuition that if Plato simply picks one of the numbers to call out then it shouldn't affect the probability of whether the sum is 7. The apparent paradox arises because you're doing the math assuming that Plato's acting as described in the first paragraph, but really he's acting as described in the second paragraph.