Yes, if I flip a fair coin 3 times I have a 1 in 2³ (i.e. 1 in 8, i.e. 12.5%) chance of throwing three heads. How do we get that result? The rule is that if P and Q are independent events, then the chance of (P and Q) = chance of P x chance of Q. Likewise, if P, Q and R are independent events, then the chance of (P and Q and R) = chance of P x chance of Q x chance of R. If each of P, Q, R as a 1 in 2 chance, then the chance of (P and Q and R) is 1 in 2³.

But, no, if I make 8 trials at throwing three heads I don't have a 100% chance of pulling it off. For the trials are independent events. And the chance of any one trial being successful is still 1 in 8, irrespective of what happened in the previous trials. Likewise, the chance of any one trial being unsuccessful is 7 in 8, irrespective of what happened the previous trials. So the chance of eight trials being unsuccessful is (7/8)⁸, which is about 0.34. So the chance of getting three heads at least once in 8 trials is .66, i.e. about two thirds (good, but very far from 100%).

Of course, the more trials at throwing three heads you make, the greater the chance of success. Thus sixteen trials will give you about a 90% chance. And the longer you go on, the nearer you get to 100%. But strictly speaking you never hit 100%: however long the sequence of trials, there remains an increasingly tiny but non-zero chance that you still fail to throw three heads every time, because the events are independent.

Yes, different notions are indeed at stake here. We need to distinguish physical probabilities from evidential probabilities.

Physical probabilities, also known as chances, are what are involved when we say, for example, that

1. An atom of plutonium 238 has a 50/50 chance of decaying within 88 years.
2. Smokers have a greater chance of getting lung cancer than non-smokers.
3. The chance of rolling 1-1 with a particular throw of a pair of fair dice is 1/36.

Note, the half-life of a plutonium atom is an objective physical property of it (a property it has independently of our beliefs about it). Likewise the probability of rolling "snake eyes" is a physical property of the chance set-up. And physical chance is related to another kind of physical property, namely the long-run frequency with which certain events turn up in a sufficient number of trials. For example, in the long-run, about 1 throw in 36 will turn up snake eyes. But philosophers argue over the relationship between the chance of a particular event and the long-run frequency in a sequence of trials.

Epistemic probabilities, by contrast, are measures of the weight of evidence. These are involved when we say, for example,

1. Given the evidence, it is very probable that human activity is responsible for the rate of global warming.
2. The absence of any large footprints shows it is very unlikely that the butler did it.
3. The evidence (or rather lack of evidence) makes it improbable that Big Foot exists.

When in a civil court, you are asked to judge "on the balance of probabilities", again you are being asked to weigh the evidence.

To help see the difference between weight of evidence and chance, note that you might well have evidence that makes it (epistemically) highly probable that a highly improbable (low-chance) event occurred. In the casino, we all seem to witness someone fairly draw the 10, J, Q, K, A of spades in sequence. A very low-chance event that on the balance of probabilities -- if we'd seen that enough checks against cheating were in place -- almost certainly happened! Note too that getting evidence for something doesn't make it more likely in the sense of increasing its chances. As we seek more evidence that you threw snake eyes three times in a row yesterday, we aren't changing the chances of that past event because it is over and done with: we can't change the physical properties of the past!

The degree to which evidence does support some conclusion is not just a matter of mere personal subjective decision. Perhaps though it is in some sense an intersubjective issue, a matter of how idealized unprejudiced rational thinkers would tend to weigh the evidence. Or perhaps not! Philosophers argue over this too.

Anyway, in sum, the two cases mentioned in the question do indeed involve two different notions of probability.

And in fact there's a third important notion of probability I should mention too, namely subjective probabilities. Subjective probabilities or credences are measures of personal degrees of confidence. Rationally or otherwise, I hold some beliefs much more confidently than others. My degree of confidence is reflected in how I behave on the basis of those beliefs. For example, I only risk crossing the road if I am very sure there is no oncoming traffic. I'll only take on a 60-40 bet that it will rain if I am that degree more confident that it will rain than that it won't. And so on.

Of course, if I'm rational and well-informed, I will apportion my degrees of belief to the available evidence, including evidence about objective chancces. My degree of belief that you will throw snake eyes on the next throw should ideally correspond to my best estimate of the objective chance of getting that outcome (if I'm overconfident, you'll be able to exploit that and extract money from me in unwise bets!). So our various notions of probability hang together more than just because chance, degree of evidential support, and degree of belief all obey (or at least under idealized conditions) the laws of the probability calculus. However, although they are linked, they are importantly different notions.

For a very lucid discussion of these things, see D.H. Mellor's short book, Probability, A Philosophical Introduction.

Thank you for your very good question. You have nothing to apologize for, and we're grateful to you for asking. I don't think I'll be able to respond to everything in what you ask, but here are a few thoughts:

Concerning the big bang, you write, "no one can say why it happened and why it did not simply stay as a singularity forever. " I should point out that even if it is true that no one can explain why it occurred, that doesn't mean there is no answer to be found. Perhaps no one can answer this question *now*, but someone (maybe you!) will someday find an answer. If that's right, then the only "randomness" here is due to our own ignorance.

Please let me mention also that contemporary physics holds that there is a very common form of indeterminacy, that is, of randomness. I mean what quantum theory has to say about the decay of an atom such as (some forms of) uranium. Whether such an atom decays at a given moment is, according to contemporary qm, entirely random. The most we can say is that there was a certain probability that it would decay within a certain period of time.

Next, I don't quite see how you are connecting randomness to the issue of free will. In fact, some philosophers concerned with free will hold that it creates its own form of free will: On this "libertarian" theory, whether I choose to perform an action or not is an event that cannot be determined by prior causes outside myself. That's a controversial view; many other philosophers will hold that freedom can occur even if the entire universe is deterministic.

So as you can see, I don't have any conclusive "answers" for you, but just some suggestions for connections. Also, if you wanto learn more, there are a lot of good things to read about this. For instance, Robert Kane has published an excellent book called _A Contemporary Introduction to Free Will_, which you may enjoy reading.

Don't stop wondering!

Good question. My own view is that what happens in the long run is irrelevant to the rationality of betting (or in your case not betting) according to the odds in the single case. I think that it is a basic principle of practical rationality that your choices should be guided by the probabilities and that, surprisingly, there is no further justification for this.

A first point. You say 'over many trials you would lose out'. Well, if you are talking about a finite number of trials, that's not guaranteed. It is possible--indeed there will be a positive probability--that in a finite number of trials you will win even if you bet against the odds. All we can say it that the probability of winning over many trials is low. So now we are just back with the original problem. Why is it rational to avoid doing something just because the probability of success is low?

Does the situation change if we think about an infinite number of trials? Well, it's not even obvious that you are guaranteed to lose if you bet against the odds an infinite number of times. Of all the infinitely many sequences of results that might happen, there's a non-empty (indeed infinite) subset of sequences on which you win in the infinite long run. True, there is a 'zero probability' of any such sequence, even thought they are all possible (that's a nice puzzle in itself). But why take that 'zero probability' to be an argument for betting against these perfect possible sequences? Once more, this looks like the puzzle we started with.



Anyway, isn't there something odd from the start about long-run justifications of betting with the odds? Why is it an argument against betting on a 'six' now that something bad will happen if I do this lots of times (or even worse if I do it infinitely many times)? In the long run we will all be dead, as Keynes said.

Suppose I am a feckless fellow with no concern for tomorrow--I need some money now. Even so it is surely rational for me to bet with the odds. But you can't persuade me of this by saying that I need to bet with the odds in order to win in the long run. I don't care about the long run.

The more I think about it, the more it seems to me that betting with the odds is a basic principle of rationality, with no further justification.

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? Third, has your hypothesis been gerry-rigged to fit the data? If so, it's no surprise that the data "confirms" it. Fourth, would your hypothesis really explain the data? Saying that the "explanation" for the facts is magic or ESP doesn't really give us much insight. Finally, how wild is the hypothesis? A low "prior probability" for the hypothesis means that it has to make the data quite a bit more likely than the alternative before the data do much to confirm it. That's why conspiracy theories are usually not credible. Ordinary human screw-ups call for far fewer moving parts.

You should distinguish here between the inductive method of extrapolating from observed cases to as yet unobserved cases, on the one hand, and particular extrapolations derived by using this method, on the other hand.

Particular extrapolations are a posteriori. They depend on what has actually been observed.

The method, however, has certain a priori elements, esp. in the very “clean” and somewhat artificial stories you will have encountered in your statistics class. One such story might be this. You are faced with a large urn which you know contains many marbles all of which you know to be either white or red. On n occasions one marble was randomly selected from the urn, its color was recorded, and it was then mixed back in. Of these randomly selected marbles, 70 percent were white and 30 percent red. At the end of the story, you are then asked what we can learn from the random drawings about the color composition of the marbles in the urn.

In this sort of story, one can calculate precisely, given the result of the drawings, the probability of various color compositions in the urn. The probabilities will peak near the ratio observed in the drawings and will concentrate in predictable ways as the number of drawings increases. (If the 7:3 ratio holds up over 1000 drawings, for instance, the probability that the real ratio is under 6:4 becomes quite small in a way that can be calculated precisely.) This is the a priori element: The rational way of adjusting one’s expectations is guided -- even determined -- by probability calculations.

In the real world, however, induction is rarely so neat. Here we need to decide what predicates are useful for extrapolation, we must worry about observations not being independent of one another, we must guard against experimenter effects and biased (theory-guided) observations, and so on.

Consider, for instance, the task of designing and fine-tuning an algorithm for accepting or rejecting mortgage applications on the basis of past repayment experience. There are indefinitely many ways of collecting and coding information about applicants. The information provided may be influenced by the conduct of the bank staff, and its coding by the bank staff’s cognitive and other biases. There isn’t just one rational way of coping with all these complexities; though some banks clearly come up with more successful algorithms than others. (Even such ex post assessments of banks are not unproblematic, however, in that only acceptance errors, nor rejection errors, will come to light. We'll never know whether the Smiths, who were denied a mortgage, would have met their debt service obligations, had they received a mortgage.)

There are a priori elements, to be sure: We can know in advance that certain features will strengthen, or weaken, a method. (For instance, a good method should work so that, the more disproportionate is the default rate of applicants with a certain characteristic, the more weight this characteristic is given as a reason to deny an application.) But much else will depend on more or less lucky guesswork and imprecise “good judgment.”

I agree that there's nothing paradoxical here; surprising, perhaps, but not paradoxical.

The only kind of additivity that is usually assumed in probability theory is countable additivity, and there's no violation of that here. But you do have uncountably many non-overlapping outcomes, each with probability zero, such that the probability of at least one of those outcomes happening is one. So uncountable additivity doesn't work.

I would agree that an outcome with probability zero need not be impossible. Consider, for example, flipping a coin infinitely many times. Each infinite sequence of heads and tails has probability zero of occurring, but one of them has to occur, so it wouldn't make sense to say that they're all impossible. (Notice that there are uncountably many possible sequences of heads and tails.)

But of course this is not a realistic experiment--no one can actually flip a coin infinitely many times. The original example proposed also seems unrealistic to me--according to the uncertainty principle, the individual protons, neutron, and electrons at the top of my head don't have precise positions, so I'm not sure it makes physical sense to think of my height as a precise real number. I'm not sure if there is any realistic example of a physical event with outcomes that are possible, but have probability zero.

Probability theorists often consider random trials with infinitely many possible outcomes each with probability zero--for example, the probability that a quantum particle will be at some particular point in space. In such cases, the probability that the result falls within an (infinite) set of such outcomes need not be zero----the probability that the particle is in some region of space, say.

I don't see that there is anything paradoxical here. It's true that cases like this violate the 'additivity' assumption that the probability of a disjunction of non-overlapping outcomes is the sum of the probabilities of the individual outcomes. But there's nothing manadatory about this assumption when we are dealing with infinite sets of outcomes, and probability theories covering this kind of case are perfectly consistent.



The question asked whether we should use the term 'impossible' for probability zero outcomes like the particle being at some particular point in space. I'd say not, given that the particle has to be somewhere. Probability zero does not mean impossible for outcomes like these.

As you make finer and finer measurements in the way you suggest, the probability declines each time by a factor of 10. As you go on and on, it shrinks below any value no matter how small. But, no matter how long you go on, it will never be zero, it will always be more than zero. (This is analogous to how, when you count, you'd eventually surpass any number anyone cares to specify but never reach infinity.) OK, so far no paradox.

But mathematics also recognizes numbers whose decimal extensions are infinitely long. And if you express each person's height as such a number, then your paradox does indeed arise. And I am not surprised, as there are other paradoxes involving infinite numbers as well, for instance, that there are as many even numbers as there are natural numbers, as demonstrable by a one-to-one mapping. Still, this is a nice addition (at least to my stock)! Let's see what others think.

Since you are obviously interested in probability and baseball, here's a fun question for you to think about. How can it happen that player A has a higher batting average than player B in the first half of the season, and A also has a higher batting average than B in the second half of the season, but B has a higher overall season batting average than A? (Yes, this can indeed happen. It is a form of "Simpson's Paradox.)