Pascal’s Pot Odds

Trust radiolab to mix poker with theology, those rascally rascals! In this episode from a couple of weeks back everything was going swimmingly with an extended discussion of poker theory, poker as philosophy, and this crazyassed hand (1) between Annie Duke and her brother Howard Lederer. In particular, they were discussing pot odds in poker. The idea that it is always worth risking a certain amount of money on a hand you may not win if the odds you are getting mean that the risk will bring rewards in the long run. In the case of four cards to a flush, for example, you have about a 1/4 chance of hitting your flush, so if there is $200 at stake in the hand it is worth calling or betting around $50. Three times out of four you’ll lose $50, but one time in four you’ll win $200 (or possibly a lot more depending on your opponent’s actions), so dividing that win by 4 you are looking at an effective profit of at least $12.50 each time you makes this play. It gets more complicated of course, but that’s the basic idea.

After the segment on poker was finished our intrepid hosts interviewed a man who had rediscovered his faith in god… using pot odds! Specifically, through reading about ‘Pascal’s Wager’. This is the argument put forward by Blaise Pascal in the 17th Century that we should call the all-in bet that is faith, because our pot odds are infinitely tempting. Because what’s at stake is eternal life or eternal damnation, says Pascal,  if we call the bet and are right we gain Heaven and avoid Hell, and if we call the bet and are wrong, well then Hell doesn’t exist so we lose nothing. If we allow Pascal his premises, then faith is like a game of poker where you are being asked to wager your life savings, but if you win you gain infinite money and if you lose you get a refund (2).

But can we grant Pascal his premises? I don’t think we can. Pascal’s wager only makes sense in a world where there are two religious options – atheism or Christianity. If the only god humanity had ever dreamt up was the trinitarian deity of the New Testament, then Pascal would be correct. But as it is, there are an inconceivable number of gods past and present floating around in the heads of us weirdos. This source gives an upper limit of 102 000 000 000 gods believed in throughout human history. This astronomical figure is based on the idea that every individual’s concept of their god is its own unique god, which is a bit tricksy for my liking. I prefer their lower estimate, based on the probable number of religious groups throughout history: about 63 000. To get an average number of gods for each religion they take the mean of 309 gods of Hinduism, the 1000 gods of the Hittite religion, and the one god of Christianity to get 436.6 gods per religion. So a good rough estimate for the number of gods through history is 436.6 x 63 000; that’s 27 510 000 gods!

So assuming that faith in each god has attendant reward, and lack of such faith has attendant punishment our pot odds start to look just a little bit worse. Now your odds of getting your eternal reward for believing in any one god are 1 in 27 509 999; you have a 0.00003635041% chance of winning this ‘hand’. When you take into account all the very rational reasons for disbelieving in any particular god, like the ‘problem of evil‘ for Christianity (3), the odds start to look even worse.

So rather than getting infinitely good odds, as Pascal asserts, we are getting odds that are inconceivably bad (4). The intellectual equivalent of betting your life saving on a horse with only three legs and a vicious case of syphilis. Considering that taking up a faith is not a matter of merely mouthing the right phrase, but involves committing your life and mind to an essentially irrational creed it seems that the safe bet is to not be an arsehole, not believe anything just because some old priest or philosopher says you should, and play shitloads of poker. When it comes to choosing one irrational faith over another, just fold the hand and find a better game.


(1) The relevant action starts around 2:20, for those interested.

(2) Funnily enough I was in a situation like this in a real poker game – a rather drunk opponent offered to pay me my buy-in plus winnings so far if I shoved all in pre-flop on at least on hand before the end of the night – whether I won or lost the hand! Kind of unbeatable odds I thought.

(3) There are lots of other reasons not to like Pascal’s theory – not least that it reduces a huge metaphysical leap of faith to a cynical calculation – but in this article I am defending the honour of poker as much as anything, so I’m not bothering with those.

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