That's an excellent question. Here is a rough reply. Oftentimes, when we refer to some everyday phenomenon as "random", we mean that we are ignorant of the fundamental causes at work -- as in games of chance. However, according to modern physics, there are some fundamental phenomena involving the behavior of sub-atomic particles that are genuinely random. For example, if a radioactive atom existing now has a half-life of (let's say) 100 seconds, then there is a 50% chance that it will decay sometime during the next 100 seconds, and there is no feature that the atom has now (or that anything else has now) that determines whether the atom will decay or won't decay. It is an irreducibly random process. In other words, the atoms that ultimately do decay before 100 seconds have passed are no different now from the atoms that do not decay during that interval. There are no "hidden variables" to distinguish them.

I should add that the reason we have for believing that these phenomena are genuinely random is NOT that scientists have looked very hard for the hidden variables, but they have not found any, and since scientists are pretty smart and resourceful, there must be no hidden variables to find. That would not be a very powerful argument! Instead, the argument is more like this: Our most empirically accurate theory of the fundamental processes of nature (namely, the theory known as "quantum mechanics") makes various statistical predictions (such as that the given atom has a 50% chance of decaying during the next 100 seconds), these statistical predictions have turned out to be quite accurate indeed, and (here's the kicker) there is no theory that hypothesizes we--behaved "hidden variables" and that makes the same statistical predictions as quantum mechanics. It's not just that no one has ever managed to come up with such a theory. It's that there cannot exist such a theory. (I haven't given you an argument for that claim -- and it should not seem obvious! Nor have I defined just what "well-behaved" means.) So either the statistical predictions made by quantum mechanics are not really as accurate as the overwhelming evidence seems to suggest, or the world is fundamentally chancy.

By the way, these sorts of fundamental chances probably have little if anything to do with the behavior of coins, dice, and so forth. At the level of such "large" objections, those chances (given the details of how the coin was tossed, the current states of the air molecules in the room, and so forth) differ only negligibly from 0% and 100%.

For a related discussion of the notion of randomness see Question 264.

Interesting points. I take it that the most reasonable reply for a defender of the ontological argument to make is to claim that Prefoessor Smith's world is not in fact possible. If one can make a case for abstracta (properties or propositions necessarily existing) then there cannot be a world where only a single pencil exists. For a good case for such a Platonic position, see Roderick Chisholm's Person and Object. R.M. Adams also has a good discussion of the difficulty of imagining / conceiving of God's non-existence. I take this up in a modest book: Philosophy of Religion: A Beginner's Guide (Oneworld Press, Oxford) or in more detail in a discussion of Hume and necessity in Evidence and Faith: Philosophy and religion since the seventeenth century (Cambridge University Press).

I confess I don't understand the notion of "metaphysical necessity," if it does not entail that that there is no possible world in which the "metaphysically necessary" being does not exist. But only a pencil exists in world W. So I really don't see what is gained (or why the very question of God's existence is not simply begged) by the claim that God is a (metaphysically) necessary being.

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: suppose X is a proposition that, if true, is a necessary truth. I may still find myself inclined to make probabilistic claims. For example: Jones offers a mathematical conjecture. I realize that if it's true, it's necessarily true. But I'm not sure if it's true. I might say things like "I think it's probable that Jones's conjecture is true." Indeed, I believe that mathematicians really do talk and think this way sometimes.

Superficially, it's not hard to see what we should say here. We can make a distinction between "objective" probability -- which has to do with the things themselves, to to speak, and epistemic or subjective probability, which has to do with our degrees of belief. Something might have an objective probability of one, but I might not know this, and so my degree of belief might be less than one. In particular, if God actually exists, and exists necessarily, or if Jones's conjecture really is true and necessarily so, I might still have doubts, and so my degree of belief would be less than 1.

All of this is fine, though there are some remaining puzzles. Subjective probabilities are supposed to conform to the rules of the probability calculus. If the necessary truth is a logical truth, then we have a problem: logical truths have to get probability one, but there seem to be plenty of cases where we aren't sure whether something is a logical truth.

However that problem gets sorted out (and there are various possible strategies) it's not directly an issue for the case of God. We've conceded that "God exists" isn't a logical truth, but allowed that it might nonetheless be metaphysically necessary. And that means there is no straightforward conflict with the rules of probability. Whether there are subtler issues about assigning subjective probabilities of less than one to metaphysically necessary propositions is something I'll leave to people who have thought about it more than I have.

If "God exists" is necessary, then the probability that God exists is 1. Full stop. It is not either 1 or 0, it is simply 1. It is also not 0.2 or any other number.

Nothing like begging the question big-time, eh?

On the other hand, I can't see why anyone serious about the question of God's existence (even theists, who would like the answer to be affirmative, but presumably not on foolish grounds) would accept the claim that "God exists" is necessary. If that were true than the could be no possible world (=a world that can be described without contradiction) in which God did not exist. But it seems obvious that there can be such a world. Consider this description:

World W = a world in which only a single pencil exists.

It's hard to spot the contradiction in that simple world! It would be a pretty boring place to be...but wait! If anyone were to be there, it would be a different world! Whew!

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.