Alex's Adventures in Numberland - Alex Bellos [129]
Pascal was one of the first thinkers to exploit the idea of expected value. His mind, though, was occupied by much higher thoughts than the financial benefits of the dicing table. He wanted to know whether it was worth placing a wager on the existence of God.
Imagine, Pascal wrote, gambling on God’s existence. According to Pascal, the expected value of such a wager can be calculated by the following equation:
(chance of God existing)×(what you win if He exists) + (chance of God not existing)×(what you win if He doesn’t exist)
So, say the chances of God existing are 50:50; that is, the probability of God’s existence is . If you believe in God, what can you expect to get out of this bet? The formula becomes:
( × eternal happiness)+(× nothing) = eternal happiness
In other words, betting on God’s existence is a very good bet because the reward is so fantastic. The arithmetic works out because half of nothing is nothing, but half of something infinite is also infinite. Likewise, if the chance of God existing is only a hundredth, the formula is:
( ×eternal happiness) + ( ×nothing) = eternal happiness
Again, the rewards of believing that God exists are equally phenomenal, since one hundredth of something infinite is still infinite. It follows that, however minuscule the chance of God existing, provided that chance is not zero, if you believe in God, the gamble of believing will bring an infinite return. We have come down a complicated route and reached a very obvious conclusion. Of course Christians will gamble that God exists.
Pascal was more concerned with what happens if one doesn’t believe in God. In such cases, is it a good gamble to bet on whether God exists? If we assume the chance of God not existing is 50:50, the equation now becomes:
( × eternal damnation)+(× nothing) = eternal damnation
The expected outcome becomes an eternity in Hell, which looks like a terrible bet. Again, if the chance of God existing is only a hundredth, the equation is similarly bleak for non-believers. If there is any chance at all of God existing, for the non-believer the expected value of the gamble is always infinitely bad.
The above argument is known as Pascal’s Wager. It can be summarized as follows: if there is the slightest probability that God exists, it is overwhelmingly worthwhile to beliee in Him. This is because if God doesn’t exist a non-believer has nothing to lose, but if He does exist a non-believer has everything to lose. It’s a no-brainer. Be a Christian, go on, you might as well.
Upon closer examination, though, Pascal’s argument, of course, doesn’t work. For a start, he is only considering the option of believing in a Christian God. What about the gods of any other religion, or even of made-up religions? Imagine that, in the afterlife, a cat made of green cheese will determine whether we go to Heaven or Hell. Although this isn’t very likely, it’s still a possibility. By Pascal’s argument, it is worthwhile to believe that this cat made from green cheese exists, which is, of course, absurd.
There are other problems with Pascal’s Wager that are more instructive to the mathematics of probability. When we say that there is a 1-in-6 chance of a die landing on a six, we do this because we know that there is a six marked on the die. For us to be able to understand in mathematical terms the statement that there is a 1 in anything chance of God existing, there must be a possible world where God does in fact exist. In other words, the premise of the argument presupposes that, somewhere, God exists. Not only would a non-believer refuse to accept this premise, but it shows that Pascal’s thinking is self-servingly circular.
Despite Pascal’s devout intentions, his legacy is less sacred than it is profane. Expected value is the core concept of the hugely profitable gambling industry. Some historians also attribute to Pascal the invention of the roulette wheel. Whether