Pascal’s Wager 101

Did you know that Blaise Pascal’s most important contribution to the theory of probability is the notion of expectation value? Briefly, the expected value of any given wager is the value of the prize multiplied by the probability of winning the prize. The earliest, and perhaps the most ingenious, illustration of this idea appears in Pascal’s quixotic philosophical treatise, Pensées (Penguin edition, pp. 123-124). In one of his essays, Pascal offers a rational argument (as opposed to purely religious or faith-based reasons) for believing in God.

In summary, Pascal’s famous wager can be expressed mathematically as follows: consider a strategy set (S) containing two choices: one can either believe in God (S1) or be an Atheist (S2). Now, let p be the probability that God exists, and 1 – p the probability that God does not exist. In essence, belief in God is like a bet or a coin toss, and given the structure of this game, there are four possible scenarios or outcomes:

(1) You choose strategy S1 (i.e., you bet on God’s existence), and God does, in fact, exist. Under this happy scenario, you will be rewarded handsomely with eternal life in Heaven and infinite joy and happiness. We further assume that the value of this heavenly reward approaches infinity (∞).

(2) You choose strategy S1, but it turns out that there is no God. Given this state of affairs, you obtain a negative payoff –c. We assume that c is a negative value, lesser than 0, because of all the lost time one may have wasted during one’s lifetime praying to a non-existent God.

(3) You choose strategy S2 (i.e., you bet against God, so to speak), but it turns out that God does exist, after all. Under these circumstances, you will be punished with eternal damnation and misery for betting against God, with a value of negative infinity (–∞).

(4) You choose strategy S2 and there is no God. Given these conditions, one will obtain a reward b in which b is greater than 0 but less than infinity (0 < b < ∞). We assume that b has a positive value, since instead of wasting your time on prayers, one was able to engage in other, more productive pursuits.

Given these payoffs, we can express the expectation value (EV) of believing in God as follows:

EV(S1) = (p)(∞) + (1 – p)( –c)

And similarly, the expectation value of the Atheist is as follows:

EV(S2) = (p)( –∞) + (1 – p)(b)

If p is zero, then it pays to be an Atheist, since b is greater than –c. The problem, however, is that no one knows with mathematical certainty whether there is a God, and even if the probability that God exists is miniscule, the expected value of believing in God is infinity. Therefore, according to the logic Pascal’s Wager, the safe or rational bet is to believe in God.

Of course, there is just one small problem with Pascal’s brilliant analysis–it assumes there is just one true God. But what if the Christian God is not the same as the Jewish God or Moslem one? What if, in short, there is more than one possible God? Which one should you worship or believe in?

When I’m not blogging, I am a business law professor at the University of Central Florida.
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5 Responses to Pascal’s Wager 101

1. shelbynfla says:

Maybe 1 does equal 2 at least… Although… if Math is finite and the law and illegalities creep into everything… And Law is absolute in that everything which usually depends on other things… then… can maybe 1 can equal 2 ? (even if it is illegal to divide by 0, technically…)
1=2:
Step 1: Let a=b.
Step 2: Then a^2 = ab,
Step 3: a^2 + a^2 = a^2 + ab,
Step 4: 2 a^2 = a^2 + ab,
Step 5: 2 a^2 – 2 ab = a^2 + ab – 2 ab,
Step 6: and 2 a^2 – 2 ab = a^2 – ab.
Step 7: This can be written as 2 (a^2 – a b) = 1 (a^2 – a b),
Step 8: and cancelling the (a^2 – ab) from both sides gives 1=2.

Hopefully we do get to find out someday who is actually right!

2. enrique says:

That’s a hard question, to say the least. Is mathematics just a cultural artifact, like art and language, or are there universal and timeless “mathematical truths”? If the latter, is this support for the existence of God? Perhaps! Maybe God is just a number!!!