Lloyd: What do you think the chances are of a guy like you and a girl like me ... ending up together? … What are my cha

1. When you have eliminated the impossible, whatever remains, however improbable, must be the truth. Sherlock Holmes, The Sign of Four [Barbossa is about to kill Will, but Jack shows up.]Barbossa: It's not possible!Jack Sparrow: Not probable. Pirates of the Carribean [Ali G, interviewing the Surgeon General C. Everett Koop] Ali G: So what is the chances that me will eventually die? C. Everett Koop: That you will die? - 100%. I can guarantee that 100%: you will die. Ali G: You is being a bit of a pessimist. Lloyd: What do you think the chances are of a guy like you and a girl like me ... ending up together? … What are my chances? Mary: Not good. Lloyd: You mean, not good like one out of a hundred? Mary: I'd say more like one out of a million. Lloyd: So you're telling me there's a chance … YEAH! Dumb and Dumber

2. A plea for the improbable Alan Hájek

3. The improbable in science, philosophy, and elsewhere I will begin my case for the importance of improbable events—my plea for the improbable—by gesturing at a number of areas in which they have found application.

4. Null hypothesis H0 vs alternative hypothesis H1. Reject H0 if, by its lights, data at least as extreme as that observed is too improbable (typically less than 0.05 or 0.01). ‘Improbable’ must be relativized to a probability function. Statistical significance testing

5. As well as data fitting a given hypothesis too poorly, it can also fit it suspiciously well. Data ‘too good to be true’

6. Fisher accused Mendel of cooking the books in the published results of his pea experiment. The probability that by chance alone the data would fit Mendel’s theory of heredity that well was 0.00003. Data ‘too good to be true’

7. Cournot’s principle:an event of small probability singled out in advance will not happen. Cournot’s Principle

8. It was advocated by Kolmogorov. Cournot’s Principle • And Borel: “The principle that an event with very small probability will not happen is the only law of chance.”

9. The principle still has some currency, having been recently rehabilitated and advocated by Shafer. Cournot’s Principle

10. But I think that Aristotle got it right: “It is likely that unlikely things should sometimes happen.” Cournot’s Principle

11. Cheney’s Principle • “If there’s a 1% chance that Pakistani scientists are helping Al Qaeda build or develop a nuclear weapon, we have to treat it as a certainty in terms of our response.”

12. Cheney’s Principle • USA had to confront a new kind of threat, that of a “low-probability, high-impact event”.

13. Cournot effectively rounds down the event’s low probability, treating it as if it’s 0. Cheney’s Principle

14. Cournot effectively rounds down the event’s low probability, treating it as if it’s 0. Cheney rounds it up, treating it as if it’s 1. Cheney’s Principle

15. Jet Propulsion Laboratory: the probability of launch failure of the Cassini spacecraft (mission to Saturn) was 1.1 x 10-3. According to the US Nuclear Regulatory Commission, the probability of a severe reactor accident in one year is imprecise: [1.1 x 10-6 , 1.1 x 10-5] Engineering and risky events

16. Improbable events have earned their keep in philosophy. A concern with improbable events has driven philosophical positions and insights. The improbable in philosophy

17. The lottery paradox puts pressure on the ‘Lockean thesis’: belief is sufficiently high degree of belief. The lottery paradox

18. For a given threshold (say, 0.95) consider a sufficiently large lottery (say, 100 tickets). Your degree of belief that ticket #1 will lose > 0.95 By the Lockean thesis (with that threshold), you believe that ticket #1 will lose. Likewise for ticket #2, …, ticket #100. But you believe that some ticket (of #1, … ,#100) will win. You have inconsistent beliefs! The lottery paradox

19. For the Lockean thesis to be plausible, the putative threshold for belief must be high. Accordingly, the probabilities involved in the lottery paradox will be small. The lottery paradox

20. For the Lockean thesis to be plausible, the putative threshold for belief must be high. Accordingly, the probabilities involved in the lottery paradox will be small. Lotteries cast doubt on Cournot’s principle. The lottery paradox

21. For the Lockean thesis to be plausible, the putative threshold for belief must be high. Accordingly, the probabilities involved in the lottery paradox will be small. Lotteries cast doubt on Cournot’s principle. We see an important feature of small probabilities: they may accumulate, combining to yield large probabilities. The lottery paradox

22. Vogel: I know where my car is parked right now. But I don’t know that I am not one of the unlucky people whose car has been stolen during the last few hours. Skepticism about knowledge

23. I have pointed out many ways in which scientists and philosophers do care about the improbable. This does much to build my case that they should care… Why care about the improbable?

24. There are specific problems that arise only in virtue of improbability. We want a fully general probability theory that can handle them. We want a fully general philosophy of probability. Why care about the improbable?

25. Probability interacts with other things that we care about, and something being improbable can matter to these other things. Why care about the improbable?

26. There are problems created by improbable events that are similarly created by higher probability events; but when they are improbable we are liable to neglect them. Skepticism about knowledge (as Vogel argued) Counterfactuals (as I will argue) Why care about the improbable?

27. I will count as improbable these propositions (‘events’): 1. Those that have probability 0. 2. Those that have infinitesimal probability—positive, but smaller than every positive real number. 3. Those that have small positive real-valued probability. ‘Small’ is vague and context-dependent, but we know clear cases when we see them, and my cases will be clear. 4. Those that have imprecise probability, with an improbable upper limit (as above). What is ‘improbable’?

28. There are various peculiar properties of low probabilities. I want to use them to do some philosophical work. I will go through these properties systematically, showcasing each with a philosophical application … What is ‘improbable’?

29. Much of what’s philosophically interesting about probability 0 events derives from interesting facts about the arithmetic of 0. Each of its idiosyncrasies motivates a deep philosophical problem. Probability 0events

30. To be sure, we could reasonably dismiss probability zero events as 'don't cares' if we could be assured that all probability functions of interest assign 0 only to impossibilities—i.e. they are regular/strictly coherent/open-minded. Open-mindedness

31. Open-mindedness is part of the folk concept of probability: ‘if it can happen, then it has some chance of happening’. Open-mindedness

32. Open-mindedness has support from some weighty philosophical figures (e.g. Lewis). Open-mindedness • We will see how much trouble probability-zero-but-possible events cause. It would be nice to banish them!

33. There are apparently events that have probability 0, but that can happen. Closing open-mindedness?

34. A fair coin is tossed infinitely many times. The probability that it lands heads every time HHH … is 0. We will revisit this claim later, but assume it for now. Closing open-mindedness?

35. A dart is thrown at random at the [0, 1] interval… Closing open-mindedness?

36. Closing open-mindedness? 0 1 • Various non-empty subsets get assigned probability 0: • All the singletons • Indeed, all the finite subsets • Indeed, all the countable subsets • Even various uncountable subsets

37. So there are various non-trivial and interesting examples of probability0 events. They create various philosophical problems, each associated with a peculiar property of the arithmetic of 0. Probability 0 events

38. The ratio analysis of conditional probability: … provided P(B) > 0 You can’t divide by 0: problems for the conditional probability ratio formula

39. What is the probability that the coin lands heads on every toss, given that the coin lands heads on every toss? You can’t divide by 0: problems for the conditional probability ratio formula

40. What is the probability that the coin lands heads on every toss, given that the coin lands heads on every toss? 1, surely! You can’t divide by 0: problems for the conditional probability ratio formula

41. What is the probability that the coin lands heads on every toss, given that the coin lands heads on every toss? 1, surely! But the ratio formula cannot deliver that result, because P(coin lands heads on every toss) = 0. You can’t divide by 0: problems for the conditional probability ratio formula

42. There are less trivial examples, too. You can’t divide by 0: problems for the conditional probability ratio formula

43. There are less trivial examples, too. The probability that the coin lands heads on every toss, given that it lands heads on the second, third, fourth, … tosses is ½. You can’t divide by 0: problems for the conditional probability ratio formula

44. There are less trivial examples, too. The probability that the coin lands heads on every toss, given that it lands heads on the second, third, fourth, … tosses is ½. Again, the ratio formula cannot say this. You can’t divide by 0: problems for the conditional probability ratio formula

45. The zero-probability problem for the conditional probability formula quickly becomes a problem for the updating rule of conditionalization, which is defined in terms of it: Suppose that your degrees of belief are initially given by Pinitial (–), and that you become certain of a piece of evidence E. Pnew(X) = Pinitial(X | E) (provided Pinitial (E) > 0) Trouble for conditionalization

46. Suppose you learn that the coin landed heads on every toss after the first. What should be your new probability that the coin landed heads on every toss? Trouble for conditionalization

47. Suppose you learn that the coin landed heads on every toss after the first. What should be your new probability that the coin landed heads on every toss? ½, surely. Trouble for conditionalization

48. Suppose you learn that the coin landed heads on every toss after the first. What should be your new probability that the coin landed heads on every toss? ½, surely. But Pinitial(heads every toss | heads every toss after first) is undefined, so conditionalization (so defined) cannot give you this advice. Trouble for conditionalization

49. To be sure, there are some more sophisticated methods for solving these problems. Trouble for conditionalization

50. To be sure, there are some more sophisticated methods for solving these problems. Various authors have written on this topic, including myself. Trouble for conditionalization