Hang out with philosophers or theologians long enough, you're likely to run into "Pascal's Wager": the Blaise Pascal's idea that you should believe in God, because if He exists, betting on Him wins you everything and betting against Him loses you everything, whereas if He doesn't, you lose nothing.
Right off the bat, we can see this original version of the wager is an intellectually dishonest argument: you don't "lose nothing" if you choose to believe that God exists and He doesn't. At best, you're being credulous; at worst, if you're being cynical about your belief, you're sacrificing your intellectual integrity.
Pascal backs off from all or nothing a bit as he's trying to dig himself out of the hole, claiming that he's comparing infinite gains of eternity in heaven against finite losses you can experience here on Earth. Some may have sincere trouble in believing, but he argues they should try to convince themselves.
Now, let's be fair to Pascal here: if you read his original text, he wasn't actually trying to convince atheists to believe per se, but instead, trying to show that the world is too uncertain for logical proofs of the existence of God, but we're probably better off acting like God exists, in case it moves us to faith.
Unfortunately, Pascal died before he could fully explain himself: the wager appears to be the introduction of a book on the value of faith that he never finished. But, like a philosophical zombie, the argument has continued its life, hollowed out from its original intent, eating brains in every new generation.
Let's slay this zombie, shall we?
Pascal's wager first appears to be an exercise in game theory: a mathematical formalism for analyzing the best choices in games. In this case, you are playing a game against the Cosmos. Your move is to believe, or not, and the Cosmos's "move" is whether God exists, or not.
[Now, the theologically savvy among you might feel like pointing out that God created Creation, and is not a part of it - which is why I used Carl Sagan's more inclusive formulation of the Cosmos as "all that is, was, and ever shall be," and I'm going to run you off with a broom if you argue about what "is" means].
This leads to a simple table: your choice of belief times the existence of God. If He is, and you choose to believe: payout plus infinity; choose not to believe: payout minus infinity. If He is not, whether you choose to believe or not, the payout is zero, or at least finite. Pick the cell with the highest value.
The emotional force of this argument is strong - for the believer - for, in decision theory, we should weigh the probability of one cell against the other, and intuitively, unless we judge the possibility of God to be literally zero, the infinite payout of the God-exists column dominates finite payouts of God-doesn't.
Mathematically, that's, um, specious at best - it looks true, but it's not a valid decision-theoretic argument. First off, Pascal put infinity in the God column specifically to outweigh any possible finite payout, but technically, we can't multiply infinite quantities by finite quantities this way.
Now, when it comes down to the question of whether infinities are actually real, or just a bad metaphor that leads people astray, I'm firmly ready to go to infinity - and beyond! But, technically mathematically, most of the time "infinity" is just a stand in for "this process can go on indefinitely without a limit."
As soon as you admit that the payout of Heaven might be finite for the purposes of modeling, then the probability assigned to the "God exists" column can be set so low that the "God doesn't" column becomes attractive. But that gets us no further than Pascal and his strict (zero-probability) unbelievers.
To me, the key flaw in Pascal's wager is what physicist E. T. Jaynes called the "mind projection fallacy": assuming that the constructs you're using in your mental models exist in reality. That's how Pascal can even put the wager to someone in the first place: he sets up the board and says "you must wager".
But the gameboard Pascal sets up doesn't exist in reality, and there's no reason for someone else to model the problem the same way. A student of religion might add columns for different views of God: Jesus who saves, Zeus who's a jerk, the Great Electron, which doesn't judge, but just is, whoa whoa.
Equally well, a student of epistemology might add many columns for belief: strict disbelief, partial belief, certain belief; an evangelical might add columns for "the hope so's" and "the know so's". Even the probabilities of columns are up for grabs. We've got a matrix of confusing possibilities.
This flaw in the wager, like the flaws in much science and folk psychology about belief, is that we do not reason about facts provided by others according to the models in the other's head: we reason about the claims that others make about facts, which we internalize based on own beliefs - and trust of the other.
Even in the simplest form, moment you start counting the columns of the wager as beliefs, the infinities disappear: there's only a claim of infinite goods in heaven, and a claim of infinite punishment in hell - and a claim that the alternative yields you only finite rewards.
And those claims are mixed in with everything else we know. As a mathematical exercise, the self-contained four-cell version of the wager has a maximum payout in the "believe in a God who exists" cell; as something that corresponds to reality, the cells of the wager start to leak.
Mathematics is an abstraction of reality - an act of creative human imagination to create repeatable forms of reasoning. I'm on the side that there is an actual reality behind this repeatability of mathematics, or it would not work; but applying mathematics to any particular problem must leave out certain details.
This is leads to the law of leaky abstractions: the notion that, no matter how good the abstraction, sooner or later it is going to fail to model the world. Forget game theory, decision matrices, and probabilities: even something as simple as the mathematical concept of number can break down.
One of the reasons I haven't published my tabbouleh recipe is that it's hard to quantify the ingredients - two bunches of parsley, four bunches of scallions, six tomatoes, two cups of fine bulgur, the juice of a lemon, etc - but since tomatoes are of different sizes, that "six" is a messy number.
But at least tomatoes come in integral quantities. Parsley comes in bunches, which are not just of different sizes; they're composed of individual stems, picked from different plants, which have different degrees of growth, freshness and wilt. Parsley needs to be cleaned and picked to use in tabbouleh.
Sometimes, you need to buy three bunches of parsley in order to end up with two. That's the law of leaky abstractions for you: you have to purchase parsley in integral units of bunches, but the bunches themselves don't correspond to the quantities that you can actually use in your recipe.
Picking beliefs for use in our minds is far more complicated than assembling a heritage Lebanese salad. There are thousands of potential facts affecting any given problem, more intertwined than the branching leaves of those leafy greens; but like them, some are fresh and edible, others black and wilted.
This was the actual point of Pascal's argument, the one he hoped to expound on his unfinished book. But the wager, because it's a mathematical abstraction - because it's repeatable reasoning - has lived on, a zombie argument which purports to give a rational reason why you should believe in God.
Ultimately, we need to carefully winnow through information that we get from others before incorporating it into our beliefs; there is no royal road to convincing anyone of anything, much less God. As for belief in God, many Christians think that must ultimately come not from reason, but from grace.
Fortunately, God gives that gift of belief for free, if we want it.
Pictured: Blaise Pascal.