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JM Keynes wrote on fundamental uncertainty that for some events in the future (such as whether or not there would be another European war or the interest rates 20 years from), we simply do not know what will happen. This is to say that there is no probability distribution at all - just complete uncertainty. Is this a coherent statement? It seems that there is always a probability for any given scenario (even if it the variables are extremely complicated). Chaos theory also seems to tell us that in a deterministic world there are some events that are too complex to predict. Are these not just a result of a lack of data or, perhaps, mathematical technique?

It depends on what you think probability is, but even then the answer is probably (heh!) no. Nothing in the mathematical theory of probability requires that all events have probabilities. Probability theory simply imposes coherence conditions on any probability assignments there may be. And the mathematical theory of probability doesn't tell us what probability is but only what its formal properties are. Some believe that there are objective probabilities—that if we specify our probability question appropriately, then there may be an answer to the question independent of what anyone thinks. For example: someone might think that if a quantum system has been prepared in a certain way, then the probability that a measurement interaction will have a certain result is, say, 1/3 regardless of what anyone thinks. This may or may not be right, though it still leaves us in the dark about what exactly this probability is. Is it a propensity or tendency of some sort? Is it a disguised way of talking about...

Consider a machine that generates numbers at random. Let's say it generates the number 12. Is there is a reason why 12 was selected rather than another random number?

Let's suppose that the machine is my computer and I'm using the function =TRUNC(100*RAND(),0). Then as I put the function in more and more cells, I'll get a list of integers between 0 and 100 that pass various tests for randomness. Let's suppose that the fifth integer on the list is 12. Is there a reason for that? There is, at least superficially. The function =TRUNC(100*RAND(),0) works by performing various well-defined mathematical operations on an input. The input is the time when you hit "ENTER," according to the computer's clock. Given that input and the cell, the output is determined. Put another way, if two computers ran the program starting at the same time according to their clocks, they would give the same output. So there's an explanation for why the fifth cell ends up containing 12 rather than some other integer. It's a matter of the input and the program. You might protest that this isn't truly random. If it were, two computers with the same input wouldn't produce the same supposedly ...

Quantum mechanics seems to suggest that there really is such a thing as a random number, yet all of philosophy and logic point to a reason or cause for everything, perhaps beyond our understanding. Is this notion of a random number just another demonstration of limited human understanding?

I guess I'd have to disagree with the idea that "all of philosophy and logic point to a reason or cause for everything." There's certainly no argument from logic as such; it's perfectly consistent to say that some events are genuinely random. Some philosophers have held that there's a reason (not necessarily a cause in the physical sense, BTW) for everything, but the arguments are not very good. On the other hand... quantum mechanics is a remarkably well-confirmed physical theory that, at least as standardly interpreted, gives us excellent reason to think that some things happen one way rather than another with no reason or cause for which way they turned out. An example: suppose we send a photon (a quantum of light) through a polarizing filter pointed in the vertical direction. We let the photon travel to a second polarizing filter, oriented at 45 degrees to the vertical. Quantum theory as usually understood says that there's a 50% chance that the photon will pass this filter and a 50% chance that it...

Here's a probability question I've been wondering. Suppose there's a company that has a million customers. It is known that 55% of these customers are male and 45% of customers are female. Task is to guess the sex of the next 100 (of the existing) customers who are going to visit the company. For every right guess point is awarded. What's the best strategy to get most correct answers? If we consider the customers one by one, it is good plan to always guess the most probable answer and therefore guess that all 100 of the customers are male. However if we take the hundred people as a group, isn't this task analoguous to situation where one litre of seawater in a container has same salinity as seawater in general? Therefore we could guess that there are 55 males and 45 females among the group of 100 customers. Certainly, if instead of 100 people we would take the whole million customers as a group then 55%/45% split would be the true and correct answer. My question is this: what changes the way of thinking...

What you say about the individual problems is right: if I get a point for each right answer, then each time someone comes to the site, the best strategy is to guess that it's a man. (At least this is right if knowing the sex of an individual customer doesn't help predict whether s/he will visit the site or not.) This is the best strategy because if each individual visit is like a random selection of a customer from the population, the chance is greater that the selected customer will be a man. The analogy with seawater is problematic. After all, if I pick one customer, that customer won't be 55% male and 45% female. The salinity of small samples of seawater closely approximates the salinity of the sea (unless we get down to really small samples of a few molecules, and then your principle breaks down.) The make-up of a small sample from a population may depart markedly from the make-up of the populations. What's interesting is that once our samples get to be of even a moderate size, things...

In a chapter on regression to the mean (Thinking Fast and Slow) Daniel Kahneman resorts to "luck" as an explanation for why one professional golfer shoots a lower score in a round than his/her rivals given that the talent pool is reasonably even. While a "lucky" (or unlucky) bounce can impact one's score, I find luck as a concept a poor explanation for performance. What is the philosophical status of luck, and are there different flavors of luck depending upon the philosophy? Is luck to chance as evidence is to data?

Games typically involve a blend of things that a player can control and things s/he can't. A golfer can work on her backswing; she can't do anything about the moment-by-moment shifts in the wind and the fine-grained condition of the greens. Things like the winds and the lay of the greens or the outcome of a dice-roll are what we might call externalities. It's not that they have no explanations and it's certainly not that they have no bearing on who wins and who loses. But the players don't deserve any blame or credit for how they turned out. In that sense, they're matters of luck. Depending on the game, skilled players may have ways of compensating for them to some extent, but they can produce advantages and disadvantages that are outside the players' control. With that in mind, I don't take Kahneman's appeal to "luck" to be an explanation. An explanation would call for specifics about conditions and causes, and the mere appeal to luck doesn't provide any of those. I take the appeal to luck to be a...

I'm going to ask a somewhat bizarre question concerning casuality, probability, and the nature of belief so bear with me thanks! Suppose a craps player goes to two casinos in Macau, the first one architecturally built according to feng shui principles and a second one not according to feng shui principles. Feng shui is an ancient Chinese system of geomancy that modern psychologists tend to discredit. This craps player personally believes in feng shui himself but only to a moderate extent. He frequents both casinos equally and bets exactly the same way every time but he usually wins at the first casino and usually loses at the second casino. 1) Does this prove that feng shui is "real," at least for him? 2) Pragmatically, even if feng shui isn't "real" or cannot be proven to be real, isn't it advisable for him to stop going to the second casino? 3) Can psychology really influence probability involving human decisions?

Statistics could give evidence that something about one of the casinos makes it more likely that your gambler will win there. Feng shui could be the explanation, though it would be a funny sort of feng shui that only worked for some of the gamblers, and so if it is feng shui, the casino may not be in business long! The more general question is whether there could be serious evidence that the gambler is more likely to win in one casino than the other, and the answer to that is yes. It might be feng shui, but other explanations, weird and mundane, would also be possible. (Maybe he's an unwitting participant in a psychology experiment; and the experimenters load the dice in his favor in one of the casinos.) Careful observation and experiment might even hone in on the explanation, if there really is a stable phenomenon to be explained. As for the pragmatic question, why not? If the evidence suggests that he's more likely to win in one casino than the other, he could go with the evidence without...

Recently, Nate Silver won acclaim by correctly predicting the electoral results for all fifty states. If one of Silver's predictions had failed, however, would that have shown that he was wrong? I mean, I take it that Silver's predictions amount to assignments of probability to different outcomes. Suppose that I claim that an ordinary coin has a 50% chance of landing head or tails. If a trial is then run in which the coin lands tails three times in a row, we wouldn't take this to mean that I was wrong. Along similar lines, then, would it not have been possible for literally all of Silver's predictions to have failed and yet still be correct?

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...

Is there such thing as coincidence? I mean is it possible that something happen without any purpose or significance?

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...

Let's say there is some crime committed and that only 5% of similar crimes are committed by someone like Person A (based on demographics, personality type, previous criminal record, etc.). If the police later find evidence suggesting that Person A is the perpetrator of a crime and that there is only a 10% chance that the evidence could exist if Person A is innocent, then does that mean there is a 90% chance that Person A is guilty? Or do we have to factor in the fact that there was only a 5% probability that A was guilty before the evidence was found? Thanks!

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...

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