Right, as Silver himself would be the first to agree. However, we might want to put it a bit differently. The projections could all be mistaken, but not because his methods or premises were incorrect. Here's a way to see the general point. Suppose we consider 20 possible independent events, and suppose that for each, the "correct" probability that the event will happen is 95%. (I use shudder quotes because there's an interesting dispute about just what "correctness" comes to for probability claims, but it's a debate we can set aside here.) Then for each individual event, it would be reasonable to project that it would occur. But given the assumption that the events are independent, the probability is over 64% that at least one of the events won't occur, and there's a finite but tiny probability (about 1 divided by 10 26 ) that none of the events will occur. So it's possible that all the projections could be reasonable and all the probabilities that ground them "correct," and yet for some or...

Suppose you and I are in the same room and we're bored. We start flipping coins. I flip twice; so do you. I get "Heads; Tails," so do you. Sounds like a meaningless coincidence to me. In fact, it would take a lot of argument to make the case that it was anything other than meaningless. Surely what's just been described is possible, and so meaningless coincidences are possible. But surely it's also the sort of thing that's actually happened countless times, and so meaningless coincidences are more than just possible. The more interesting question is whether anything has purpose or significance apart from the purpose or significance that creatures like us give it. Put another way, the question is whether there's any significance inherent in the universe itself. Many religious believers would say yes, though they would trace the meaning back to the intentions of God. Carl Jung, the Swiss psychologist, believed in meaningful coincidences that he called "synchronicity." His account of them (as I...

What we're trying to get to is the probability, given all the evidence, that A is guilty. Let H be the hypothesis that A is guilty. You're supposing that our initial probability for H is 5%, i.e., .p(H) = .05. Then we get a piece of evidence – call it E – and the probability of E assuming that H is false is 10%, i.e., p(E/not-H) = .1. Your question: in light of E, how likely is H? What's p(H/E)? We can't tell. We need another number: p(E/H). We need to know how likely the evidence is if A is guilty. And we can't infer that from p(E/not-H). Why not? Well, suppose the evidence is that the Oracle picked A's name out of a hat with 10 names, only one of which was A's. The chance of that if A is not guilty is 10%, but so is the chance if A is guilty (assuming Oracles don't really have special powers.) iI this case, the "evidence" is actually irrelevant. The crucial question is this: what's the ratio of p(E/H) to p(E/not-H)? Intuitively, does H do a good job of explaining E? And knowing only one...

I'd suggest that there needn't be a determinate answer without adding more detail. In particular, the notion of "mixing" the three collections would need to be spelled out. Suppose the "mixing" works this way: take 10 marbles from the red bag and one from each of the blue bags. Put in an infinite vat and stir. Repeat ad infinitum. (We could imagine the first operation is performed in 1 minute, the second in half a minute, the third in a 1/4 minute…) The intuitive thought is that a "random" draw is most likely to give you a red marble. (10 chances out of 12). This may seem contrived, but only because we have some other loose, unspecified idea of mixing that we're comparing it to. The point is simply that the problem, as stated, doesn't determine the answer.

I'd like to offer a rather different take on this than my co-panelist. Many theists don't think that "God exists" is a necessary proposition. However, some famously do. St. Anselm is the most well-known example, but he's not the only one. The contemporary philosopher Alvin Plantinga apparently does as well. Now we can grant that it's not obviously a contradiction to say that the world contains only a single pencil, but people who think God exists necessarily may not think that metaphysical necessity is the same as logical necessity. If I understand Plantinga correctly, he doesn't think it's a contradiction to say "God doesn't exist," though he does think that God's existence is metaphysically necessary. All of that is throat-clearing. We could make a similar point in a different way. Mathematical truths are necessary if true at all, or at least so we'll suppose. But it's famously hard to argue that mathematical truth is the same as logical truth. So the more interesting question is this:...

There's no simple answer to this question, but there is a caution: both common experience and a good deal of psychological work suggest that we have a strong tendency to project patterns onto random events. We also tend to notice things that interest us and ignore things that don't. And remember that it is overwhelming probable that some improbable events or other will occur. A single run of ten heads in a row on flipping a fair coin has a chance of 1 in 1,024. But if lots of people perform the same experiment, it becomes nearly certain that someone will get 10 heads. Still, some apparent coincidences do seem to call out for explanation. Without offering a full-blown story of how this should work, here are some thoughts. First, do you have a hypothesis in mind? Casting around blindly for an "explanation" may not get you very far. Second, would your hypothesis really make what you noticed that much less surprising? Or is what you noticed the sort of thing that might well have happened by chance anyway...