Combinatorial problems are almost always quite interesting. I think it's because most of them seem counterintuitive.

For example, "What is the number of possible orderings of a deck of 52 distinct playing cards? The answer is 52! (fifty-two factorial), which is equal to about 8.0658 × 10^67." -- Wikipedia, "Combinatorics"

John Allen Paulos has dealt with such questions in good books such as

Once Upon a Number:

Innumeracy;

Beyond Innumeracy.

You don't have to be an expert to understand his explanations.

Norm wrote:

I think it is the same wrong intuition that people use when presented with the fine-tuning argument for the existence of a designer god, the argument is that the laws and constants of physics are fine tuned and if they were different we wouldn’t exist. To which I answer okay, and what’s your point. Well there must be a designer to have just this set of laws and constants they respond, but like the two bridge hands above it is more likely that it is random.

'I now turn to another kind of principle or idea, and that is that there is no sense in calculating the probability or the chance that something happens after it happens. A lot of scientists don't even appreciate this. In fact, the first time I got into an argument over this was when I was a graduate student at Princeton, and there was a guy in the psychology department who was running rat races. I mean, he has a T-shaped thing, and the rats go, and they go to the right, and the left, and so on. And it's a general principle of psychologists that in these tests they arrange so that the odds that the things that happen happen by chance is small, in fact, less than one in twenty. That means that one in twenty of their laws is probably wrong. But the statistical ways of calculating the odds, like coin flipping if the rats were to go randomly right and left, are easy to work out. This man had designed an experiment which would show something which I do not remember, if the rats always went to the right, let's say. I can't remember exactly. He had to do a great number of tests, because, of course, they could go to the right accidentally, so to get it down to one in twenty by odds, he had to do a number of them. And it's hard to do, and he did his number. Then he found that it didn't work. They went to the right, and they went to the left, and so on. And then he noticed, most remarkably, that they alternated, first right, then left, then right, then left. And then he ran to me, and he said, "Calculate the probability for me that they should alternate, so that I can see if it is less than one in twenty." I said, "It probably is less than one in twenty, but it doesn't count." He said, "Why?" I said, "Because it doesn't make any sense to calculate after the event. You see, you found the peculiarity, and so you selected the peculiar case."
'For example, I had the most remarkable experience this evening. While coming in here, I saw license plate ANZ 912. Calculate for me, please, the odds that of all the license plates in the state of Washington I should happen to see ANZ 912. Well, it's a ridiculous thing. And, in the same way, what he must do is this: The fact that the rat directions alternate suggests the possibility that rats alternate. If he wants to test this hypothesis, one in twenty, he cannot do it from the same data that gave him the clue. He must do another experiment all over again and then see if they alternate. He did, and it didn't work.'
—Richard Phillips Feynman, in "The Meaning of It All: Thoughts of a Citizen-Scientist"

3) If the question is then asked as to what is the likelihood that another universe that is formed without input from an 'Intelligent Designer' also has a set of physical parameters conducive to life, the response should be that such a likelihood may well be very small, the precise value depending on the probability distribution functions of the physical parameters in question. However, the likelihood of your winning the lottery depends not just on the likelihood of any given combination of numbers showing up, but also on how many lottery tickets you buy. There are many billions of stars in our galaxy, and many billions of galaxies in our universe. Who is to say how many (billions? trillions? googolplexes?) of candidate universes (lottery tickets) there were/are/will be in toto?

One Good Move, I saw this on Friendly Atheist - I think your analysis is flawed.

The actual probabilities of getting each hand are:

Probability of hand 1 = (probability of a designer)x(probability he provides hand 1)+(probability of no designer)x(probability of randomly getting hand 1)

Probability of hand 2 = (probability of a designer)x(probability he provides hand 2)+(probability of no designer)x(probability of randomly getting hand 2)

The probability of a designer plus the probability of no designer add up to 1. To arrive at the answer that the probability of both hands are equal, you assumed that the probability of a designer=0 and the probability of no designer=1. You can't prove anything about God starting from the assertion that God doesn't exist. That's not a proof; that's an unproved assertion.

Now, if there's a nonzero probability God exists then a lot of discussion could ensue on what sort of hand God is more likely to provide. However, we cannot assume randomness in his choice of hand so your argument that both hands are equally likely is invalidated.

There's another way to look at this, though. Consider that you've entered a game that has the following rules:
1) You are dealt 13 cards. You pick them up and look at them.
2) if they are all spades, you win.
3) if they are not all spades, you are instantly killed before you even realize what's going on. (and incidentally lose the game).

Now imagine that you decide to play the game. You get your cards, and look at them, and - well, obviously you win, right? since you're still around! If you imagine this, that you've really gone ahead and played this game and you win it, you would probably still find yourself wondering why you were dealt A-2 of spades. Something so improbable cries out for an explanation, doesn't it? And the two possibilities that suggest themselves are (1) the deck was stacked, or (2) some many-worlds / multiverse scenario where you are one of 635,013,559,600 or more copies to receive a hand.

Here the winning hand was specified before the game was played; i.e., there is something "special" about the all-spades hand. Your attitude toward the fine-tuning problem (whether you think it's a problem or not) is likewise determined by whether you think there's something "special" about the observed values of the physical constants or not. I tend to think there is, and hence there is a problem that needs explanation, and that the multiverse family of explanations is the best. But YMMV.

nice comment, inwit. thanks!

I remember hearing something like this in a Q&A session at an ID lecture.

Suppose a fundamental constant of the universe is the example factor, E, equal to 1. If E were outside the interval

[1 - 10^-999, 1 + 10^-999]

the universe would be unable to support life. Assuming the interval is continuous, there are infinitely many life friendly constants. The probability of E being in the life supporting range, even if E could be anything from 10^-100000 to 10^100000, is infinity/infinity.

Can anybody more acquainted with the fine tune argument tell me whether this has any potential?

I'm not trying to prove here that God literally doesn't exist. I'm not particularly adept at proving negatives. I'm just drawing an analogy that I found telling. We shouldn't require any special explanation of there being a hand that turns out all spades because that is just as likely as any other hand. It only seems significant because the pattern of spades is significant to us. By analogy the fine tuning of the constants of the universe doesn't require any explanation. It is just as unlikely as any other configuration the constants might have taken. There is no need for explanation here and thus we need not invoke God as an explanation. Furthermore, even if we did need an explanation God doesn't provide a good one. God again, just serves as a placeholder for I don't know.

Norm, thanks for your comment. I see what you're saying. My atheist husband always has said "it DID happen - so it's possible - end of story."

"God again, just serves as a placeholder for I don't know."

nice one.