I think the more interesting question is what makes wildly implausible conspiracy theories seem plausible to so many. I think that part of the problem is that it takes some specialized training to be able to compute probabilities accurately, and most without that training tend to overestimate how likely it is that a complicated conspiracy really occurred, and I think that part of the problem is that many do not have adequate accounts of how one event might or might not explain another.

I don’t think, however, that these “philosophical” issues why so many find those implausible theories highly intriguing – I imagine there is are interesting sociological , psychological, and political explanation for this, and I hope that other panelists may be able to say more.

To answer this question, it may be helpful to say something about the mathematical formalism usually used in probability theory. The first step in applying probability theory to study some random process is to identify the set of all possible outcomes of the process, which is called the sample space. For example, in the case of an infinite sequence of coin flips, the sample space is the set of all infinite sequences of H's and T's (representing heads and tails). Probabilities are assigned to events, which are represented by subsets of the sample space. For example, in the case of an infinite sequence of coin flips, the set of all HT-sequences that start with H represents the event that the first coin flip was a heads, and (assuming the coin is fair) this event would have probability 1/2. The set of sequences that start with HT is a subset of the first one, and it represents the event that the first flip was heads and the second tails; it has probability 1/4.

Now, consider some infinite HT-sequence s. For any positive integer n, we can consider the set of all sequences that agree with s for the first n terms. This set contains s, and imitating the reasoning in the last paragraph we see that it represents the event that the first n coin flips come out as specified by s, which has probability 1/2ⁿ. Since {s} is a subset of every one of these sets, the event that the entire infinite sequence is exactly s must have probability less than 1/2ⁿ for every n. But that means that the event must have probability 0. So you are absolutely right that the reasoning here involves a limiting process: the probability is 0 because 1/2ⁿ approaches 0 as n approaches infinity.

With this background, it is also now easy to see the distinction between zero-probability events that are possible and those that are impossible. The event that the entire infinite sequence is s is represented by the set {s}. It has probability 0, but is possible. The event that the first flip is both a heads and also a tails is represented by the empty set (since there are no elements of the sample space that fit this description); it has probability 0 and is impossible.

Sometimes probability is a measure of our ignorance. If you give me a quarter with the instruction to hide it in one of my fisted hands while your eyes are closed (and I do as you say), then you'll not know which hand holds the coin. (I will know, I can feel it.) So you can only assign probabilities because you lack knowledge.

In other cases, probability is objective. If current physics is right, then some processes in nature are in principle unpredictable or such that their outcome is uncertain. Yes, this suggests some reference to a knower: it means that it's impossible for there to be someone who can predict or be certain about the outcome. But why should this be problematic? The fact that a black hole emits no light can be expressed by saying that black holes are invisible - and yet the fact is "intrinsic to reality," involves no essential reference to beings with eyes.

I'd like to add a little bit to what Thomas has said. Probability problems can be tricky because the answers sometimes depend on small details about exactly what procedure was followed. For example, the problem says that "you are given one envelope." Who gave you the envelope? Did the person who gave you the envelope know which envelope was which? Was he a very stingy person, who might have been more likely to give you the envelope with the smaller amount of money? If so, then the probability that you have the smaller amount might not be 1/2.

But that is clearly not the intent of the problem, so let us assume that the person who gave you the envelope flipped a coin to decide which envelope to give you. Then, as Thomas says, the probability is 1/2 that you have the small amount and 1/2 that you have the large amount. Suppose that you open your envelope and find $100 in it. You now know that the other envelope contains either $50 or $200. Do these two outcomes still have probability 1/2 each? Not necessarily; by opening your envelope you acquired new information, and that information could change the probabilities. The answer depends on what procedure was used to decide how much money to put in the envelopes.

Suppose the person who filled the envelopes used the following procedure: they chose a random integer x from 1 to 100, with each integer being chosen with probability 1/100. Then they put $x in one envelope and $2x in the other. In that case, a short calculation using the laws of conditional probability shows that, yes, the probability is 1/2 that the other envelope contains $50 and 1/2 that it contains $200. The expected value of the amount of money in the other envelope is therefore $125, so you would be well-advised to switch.

But now suppose the envelopes were filled by choosing x between 1 and 50, and putting $x in one and $2x in the other. Now, when you find $100 in your envelope, you know for sure that the other envelope contains $50, and you should not switch.

What if we don't know anything about how the envelopes were filled? Now the question is more difficult, but I would be inclined to say that probability theory cannot tell us the probability of the other envelope containing $50 or $200. Probability theory can only tell us how to compute probabilities if we have a well-defined probability distribution for the possible outcomes of the random event under consideration. The choice of this probability distribution is a matter of interpretation; it is not a purely mathematical issue. If we don't have enough information to determine the distribution, then it is not clear that there is a right answer to the probability question.

Wouldn't it be nice if mathematics could be brought down so easily! But, sorry, no cigar this time.

It is indeed true that the probability that you are holding the fat or meager envelope is 50/50. Here are the two cases:

1. If you are holding the meager envelope, then switching gets you from x to 2x for a gain of x.

2. If you are holding the fat envelope, then switching gets you from x to 1/2 x for a loss of 1/2 x.

But note that the "x" in these two cases does not signify the same amount of money. In Case 1, x is the smaller amount. In Case 2, x is the larger amount. In Case 1, your gain from switching is the smaller amount. In Case 2, your loss from switching is 1/2 the larger amount (equal to the smaller amount).

The illusion arises because, at first blush, the situation seems similar to another where someone offers to give you 50/50 odds on either doubling or halving some fixed amount of money you have. There your reasoning goes through and you are well-advised to accept. Your probability-weighted pay-off is 1.25x (a little more than you wrote).

But the problem your friend posed is different. Here you do not have a 50/50 chance of either doubling or halving some fixed amount of money. Rather, you have a 50/50 chance of either doubling a small amount of money or halving a large amount of money.

This is an excellent question. You are right: getting heads twenty times in a row is exactly as likely as, say, HTTHTHHTTTHHTHTHHTTH.

However, 20 heads is much less likely than (say) 10 heads and 10 tails. There are many more twenty-flip combinations that yield 10 heads and 10 tails than twenty-flip combinations that yield 20 heads. There are more ways to get 10 H and 10 T than to get 20 H.

So getting 20 heads is less likely (assuming the coin is fair) than getting 10 heads and 10 tails. Yet getting 20 heads is exactly as likely as getting a particular combination of 10 heads and 10 tails, such as HTTHTHHTTTHHTHTHHTTH.

Other cases of "strangeness" are a bit more difficult to diagnose. For instance, suppose a lottery is run and ticket #1729 wins. This outcome is extremely unlikely if the lottery is fair (let's suppose there are 10,000 tickets), but extremely likely if the lottery was fixed for #1729. Does this mean that after this ticket is drawn, we should conclude that the lottery was probably fixed for #1729 (and so the outcome was not so "strange" after all)?

No! In light of the winning ticket, our confidence that the lottery is fair should reflect not only the likelihood of the outcome if the lottery is fair, but also our initial confidence (before the ticket was drawn, that is) that the lottery was fair. If there was no particular reason before the ticket was drawn for us to have been suspicious that the lottery was fixed for #1729, then the outcome will do very little to persuade us that the lottery was fixed for #1729. On the other hand, if #1729 was owned by the person running the lottery, then we might have been suspicious initially that the lottery would be fixed for #1729, in which case this ticket might count as very strong evidence that the lottery was in fact fixed for #1729.

The improbability is taken to be that it came about through chance, or without someone intending it to be like that. The example is often giving of someone walking on a beach and finding a watch, never having seen one before. When she looks at it, she will not know what it is or what it does, but she is entitled to conclude that it could not have come about through chance, since it is so complex and organized. Whether that is a good argument is much debated, of course, but that is where the improbability is taken to lie.

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!