# Pascal, Poker and Pensées

If you’ve been reading this blog for a while you know I like to play poker.  I have read numerous poker books and articles over the years, and the concept of “expected value” is at the core of every one.  Expected value theory helps skilled players calculate whether a particular play will, in the long run, be profitable.

Here’s a simple example.  Suppose I’m playing Texas Hold ‘em and my hole cards are the king of hearts and the three of hearts.  The flop comes and the community cards are the ace of hearts, the four of clubs and the nine of hearts.  My opponent is in front of me and bets out for \$10 into a pot of \$25.  Everyone else folds.  What should I do?  This is a very complex question with numerous variables, but for my purposes here I want to isolate the variable of “mathematical expectation” with respect to the current pot (which, for the initiated, means I am setting aside issues of semi-bluffing, implied odds, stack size, opponent profile, fold equity, etc.).

If I narrow the question to mathematical expected value, the answer is simple.  First, I missed the flop and have nothing but a king high hand.  I assume my opponent either hit the flop (probably with a pair of aces) or had a pocket pair to begin with.  As things stand now, I am almost certainly beat.  Should I just fold and move on to the next hand?  No.  A call here has a positive expected value.  What?  How can calling a bet when you are beat have a positive expected value?  The answer lies in the probability of certain future events.  While I currently have nothing, if another heart comes I will have an ace-high flush, which will almost certainly win the pot.  With two cards coming I have approximately a 35% chance of completing my flush by the end of the hand.

Let’s do the expected value math:  The pot currently contains \$35 (the original \$25 plus my opponent’s bet).  I have a 35% chance of winning by the end of the hand.  \$35 X 0.35 = \$12.25.  This means that even though I will lose 65% of the time and win only 35% of the time, over the long run I should expect to receive a net average of \$12.25 if I call.  I only have to invest \$10 to obtain an expectation of \$12.25, which gives me an expected profit of \$2.25.    I should call.

Blaise Pascal and Pierre de Fermat jointly worked out the mathematical underpinnings of expected value theory in 1654.  This is interesting, because Pascal’s famous “Wager” is at bottom simply a special case of expected value theory.  Let’s see how.

Pascal describes the wager as follows:

“God is, or He is not.” But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? According to reason, you can do neither the one thing nor the other; according to reason, you can defend neither of the propositions.

Do not, then, reprove for error those who have made a choice; for you know nothing about it. “No, but I blame them for having made, not this choice, but a choice; for again both he who chooses heads and he who chooses tails are equally at fault, they are both in the wrong. The true course is not to wager at all.”

Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.

“That is very fine. Yes, I must wager; but I may perhaps wager too much.” Let us see. Since there is an equal risk of gain and of loss, if you had only to gain two lives, instead of one, you might still wager. But if there were three lives to gain, you would have to play (since you are under the necessity of playing), and you would be imprudent, when you are forced to play, not to chance your life to gain three at a game where there is an equal risk of loss and gain. But there is an eternity of life and happiness. And this being so, if there were an infinity of chances, of which one only would be for you, you would still be right in wagering one to win two, and you would act stupidly, being obliged to play, by refusing to stake one life against three at a game in which out of an infinity of chances there is one for you, if there were an infinity of an infinitely happy life to gain. But there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite.

Note that Pascal’s Wager is NOT a proof of the existence of God.  Indeed, the wager assumes for the sake of argument that God’s existence cannot be definitely proved.  Nevertheless, the wager demonstrates that even in a state of uncertainty (even extreme uncertainty) the rational man chooses to live his life as though God exists.

Very importantly, you cannot avoid the bet.  God exists or he does not.  Every person must choose which side of this question he comes down on.  Attempting to ignore the choice (i.e., choosing not to choose) is futile.  You cannot avoid the Wager.  Choosing not to choose does not mean you have not chosen.  It means you have wagered on “does not exist.”

If we must bet, then how should we bet?  The Wager demonstrates that only a fool would bet that God does not exist even if he is all but certain he is right.  What?  Why should a person bet on God’s existence even if he is all but certain that God does not exist?

The proposition can be expressed mathematically as an expected value analysis.  No honest person can ever say he is 100% absolutely positive God does not exist.  Even arch-atheist Richard Dawkins expresses the matter as “God probably does not exist.”  So let’s assume for the sake of argument that there is a 99.999999% chance that God does not exist.  What is the expected value of betting on “God does not exist” with respect to the afterlife?  Here there are actually two bets.  The bettor bets that God does not exist.  He also bets against the proposition that he does exist.  To find the expected value of these two bets we find their independent expected values and add them together.

Bet 1: God does not exist.  The bettor is wagering that God does not exist and that when he dies he simply ceases to be.  The value of ceasing to exist is zero.

EV = 99.999999% X 0 = 0.  In other words, if the bettor wins his bet he gains nothing.  Therefore, the expected value of the bet is zero.

Bet 2:  God exists.  The bettor is wagering that God does not exist and that the penalty of eternal suffering in hell will not be applied to him.  We will assign negative to the outcome “eternal suffering.”

EV = .000001 X – = –

So the expected value of the wager “God does not exist” is the sum of these two bets:  0+- = –

Now let us calculate the expected value of the bet “God exists.”

Bet 1: God does exist.  The bettor is wagering that God does exist and that when he dies he will have an eternal reward in heaven.  We will assign to the outcome “eternal reward in heaven.”

EV = .000001 X = .  If the bettor wins his bet he gains eternal bliss.  Therefore, the expected value of the bet is .

Bet 2:  God does not exist.  The bettor is wagering that God exists.  If he loses and the atheist is correct he simply ceases to be at death and loses nothing.

EV = 99.999999% X 0 = 0

So the expected value of the wager “God does exist” is the sum of these two bets:  EV = 0 + = .

In summary, the rational man chooses to bet “God exists” even if he believes the probability of that proposition being true is very low.  Keep in mind that I have assumed that the probability of God existing is very low ONLY for the sake of the argument.  In fact, given the various proofs of God’s existence that have been adduced over the centuries, I believe that it is far more likely than not that God does in fact exist.  This only adds force to the conclusions to be drawn from the Wager.

## 25 Replies to “Pascal, Poker and Pensées”

1. 1

Barry:

Fun post and good update on Pascal’s Wager. I came to a similar conclusion as a young man many years ago on my own, without ever reading or knowing about Pascal’s Wager. There are, however, a few built in assumptions that go somewhat unstated. Namely, even if God exists, it does not follow that: (i) I will continue to exist after this life, (ii) my existence after this life will depend on what I do in this life, (iii) there is some kind of reward/punishment system, (iv) God cares a whit about me enough to care how I live and to try to reward/punish me for living a particular way, and so on . . . All of those things have to be answered from a different source than my Wager.

Further, one could take Pascal’s Wager and apply it to any particular God, oh, I don’t know, perhaps the God of radical Islam, or perhaps a particularly vengeful God who wants his followers to inflict misery on the unbelievers, or perhaps the fickle Gods of Greek mythology. Now how should I conduct my life? How should I live, just in case this God turns out to be true?

Pascal’s Wager is an interesting tool to get people to stop and think about the rationality (or lack thereof) of wagering against God’s existence. And it might be helpful in opening someone’s mind to the fact that they should take the possibility of God’s existence seriously. I just don’t think it can go very far in telling us much about what kind of God we might be dealing with, what we should believe, how we should live, and so on.

2. 2
Mung says:

I pretty much always accepted that God exists. My problem was what to do about what that belief required of me.

3. 3
Barry Arrington says:

Eric,
I agree with the thrust of your comment and even anticipated that a comment like it would come. Boiled down, the unstated assumption of the Wager is that when we say “God exists” by “God” we mean the God described in the Bible. As long as there is a non-zero chance that that God exists the wager holds.

4. 4
Seqenenre says:

Betting on a competitor God might displease the real God more than betting on no God at all.
Therefore the atheist choice might have some merit after all.

5. 5

Barry,

I have a 35% chance of winning by the end of the hand.

I don’t think you can say you have a 35% chance of winning since you don’t know what hearts have already been distributed to the other players and will be unavailable to you.

But I think the use of expeced value is good. That is the heart of Pascal’s wager after all.

6. 6
Barry Arrington says:

The fine details of hold ‘em probability calculation are kind of beside the point of the post, but the probability calculation is correct. Yours is a common mistake. You are correct that I do not know what hearts have been dealt to the other players. Those cards are unknown and probability is based upon the proportion of known cards to unknown cards. Whether the unknown cards remain in the deck or have been dealt to the other players is itself unknown and does not affect the probability calculation.

There are 52 cards in the deck. After the flop there are 5 known cards (my 2 hole cards and the 3 community cards) and 47 unknown cards. After the turn there are 46 unknown cards. There are nine other hearts. The probability is calculated as the compliment of the opposite as follows:

1 – [(47-9)/47] * [(46-9)/46] = 1 – .80851 * .80435 = 1 – .6503 = .3496 or 35%

7. 7

Let’s see . . . I guess you’re working with only two players. 52 cards in a deck minus the 7 hearts that have been dealt and one non-heart and the opponents two hole cards which leaves 42 cards. We know 4 hearts have already been distributed which leaves 9 hearts. The opponent could have 0, 1 or 2 hearts. Which means there are 9, 8 or 7 hearts left in the deck of 42 cards. What is the probability that at least one heart will come up in the next two cards? The probability of the next card being a heart is 9 or 8 or 7 out of 42. If the next card is NOT a heart then the probability of the second card being a heart is 9, 8 or 7 out of 41. If the next card is a heart then the probability of the second card being a heart is 8, 7 or 6 out of 41.

Even answering that is not exactly the chances of winning. The opponent could have two aces in their hole. If the 4th ace and a heart are the next two cards you lose even with the flush.

Again, I like the idea of expected value but I think it’s more complicated in this case.

8. 8

Barry,

It’s not important to your main point which I think is fine. I just like the math stuff!! 🙂

9. 9
Barry Arrington says:

Jerad, no, the probability calculation remains the same whether I am playing with nine players or two players. I do NOT know whether the other nine hearts are in my opponents hand or in the deck. All I know is that they are unknown. Again, this is beside the point and I am gaveling further discussion on the fine points of hold ‘em probability theory. You are correct that if my opponent already has two aces and gets two more aces (called “runner runner” or “running” aces) he will beat me but in that case I won’t get the flush because neither of the running aces will be a heart, because the ace of hearts is already out. I could still be beat if I get a flush. If he has two aces and another ace comes and then board pairs (say with the 4 of hearts) he will have a full house and beat my flush. The probability of that happening is very very low (about 1.5%) and therefor it can be discounted as a practical matter. Or he could already have two pair, say ace-nine. In that case he has six outs (25% chance at the flop) to a full house if my flush comes. But if I get the the nut flush and the board is not paired I am certain to win.

10. 10
Mung says:

Now tell us about pot odds and implied odds 😉

If I ever hold a contest I know what to give you if you’re the winner, lol.

11. 11
Barry Arrington says:

Seqenenre, your comment does not make sense in the context of the God of classical theism. As Edward Feser writes:

“The God of classical theism is not “a god” among others, precisely because He isn’t an instance of any kind in the first place, not even a unique instance. He is beyond any genus. He is not “a being” alongside other beings and doesn’t merely “have” or participate in existence alongside all the other things that do. Rather, He just is ipsum esse subsistens or Subsistent Being Itself. He is First Cause not in the sense of being the cause that came before the second, third, fourth, fifth, etc. causes, but rather in the sense of having primal or absolutely underived causal power whereas everything else has causal power in only a derivative and thus secondary way. He is not “a person” but rather the infinite Intellect and Will of which the persons of our experience are mere faint reflections. Since He has no essence distinct from His existence which could even in principle be shared with anything else, He is not the sort of thing there could intelligibly be more than one of. And so forth. Nothing less than this could be the ultimate source of all things and thus nothing less could truly be divine.”

12. 12
StephenB says:

Interesting post, Barry. I have always loved Pascal because he is the master of contrasts and writes like a dream. Sometimes his sentences are so delicately constructed, they are almost seem like mathematical equations. While I am not with him in all things, especially his disdain for philosophy, I own him a great debt since he was one of my early role models for Christian apologetics.

Most importantly, his famous wager does, indeed, reflect the human condition. Once the subject of heaven and hell has been put on the table, one must come to terms with it. It’s the calculus that weights the prospect of finite pleasure and pain against the prospect of infinite reward and punishment. There is no way around it. It is not possible to not choose.

13. 13

Seqenenre @4:

LOL! I love it! Maybe you could write up an anti-Pascal’s Wager to the effect that the safest thing to do is take no position! 🙂

14. 14
carlg says:

Carl Grant I spent 5 minutes thinking about whether to write a rather complex post and another 20 minutes, of my not completely worthless time writing it only to be left with a blank comment area when I failed to do something, a captch? So the effort was waisted. If you are trying to convince people you are a fool and to not waist more time on this blog, you have succeeded.

15. 15
Bruce David says:

Barry:

Eric,
I agree with the thrust of your comment and even anticipated that a comment like it would come. Boiled down, the unstated assumption of the Wager is that when we say “God exists” by “God” we mean the God described in the Bible. As long as there is a non-zero chance that that God exists the wager holds.

I disagree. If God exists but He is in fact the God of Islamic fundamentalism, then the bet on a Christian version of God will land you in Muslim Hell. In order to calculate expected value, you need the probabilities of all the different possible versions of God that have a non-zero chance of existing, something which is of course impossible to come by.

For myself, I calculate that the “God described in the Bible” is in fact self-contradictory, so that the probability of its existence is zero. The self-contradiction is this: the God of the Bible is reportedly both infinitely, unconditionally loving and also judging, condemning, and punitive. Unconditional love never punishes, and certainly does not punish infinitely, (as in condemning souls to an eternity in Hell).

My wager is with a truly unconditionally loving God, as has been reported by myriad people who have had near death experiences.

16. 16
kairosfocus says:

CG, I understand how you must feel, there are oddities with all these things; hit back as instructed then post an accurate captcha. This should work. Hopefully it is not too late. Alternatively, compose in a word processing package then post, noting that the quote marks in a blockquote must be straight not smart. KF

17. 17

KF @15:

Having lost quite a few comments to the captcha system, I can assure you that, at least on my computer, hitting ‘back’ does not preserve what has been written.

CG:

Yeah, it is a pretty dumb system and poorly implemented. Rather frustrating at times. I (try to remember to) copy what I’m writing before hitting ‘Post Comment,’ and also have started filling out the math captcha before writing my comment, not after. Thanks for posting, and I hope the annoying system won’t prevent you from making an effort to post again in the future.

18. 18

Bruce David:

Your point about the possibility of other gods as it relates to Pascal’s Wager is well stated.

However,

. . . the God of the Bible is reportedly both infinitely, unconditionally loving and also judging, condemning, and punitive. Unconditional love never punishes, and certainly does not punish infinitely, (as in condemning souls to an eternity in Hell).

So it sounds to me like your problem with the God of the Bible is that you have a very skewed view both of (i) what the Bible says and (ii) what it means to be loving. Under your ‘logic’ a loving parent cannot punish a child for misbehaviors because that would mean that the parent doesn’t love the child. I don’t know if you have kids, but I can assure you this is most definitely false. Might want to rethink your position a bit . . .

19. 19
Barry Arrington says:

The captcha system gets me too. As a matter of fact, I got the math problem wrong the first time I tried to enter this comment and lost it. That is very frustrating, I know.

The alternative is to expose our combox to thousands of spambots who will overrun us very quickly.

Captcha is clearly the lesser of two weavils. Or is that “evils”? I can neve keep that expression straight.

20. 20

BA, yeah I understand why the captcha is necessary to combat spambots. The problem is that it doesn’t allow you to preserve your text. On may webpages if you get the captcha wrong, it just pops up a note on the same page and you get to try again. With UD it takes you to a new error page and everything you wrote goes into the memory hole. Anyway, I know I’m preaching to the choir. Don’t know if there is any other way to implement the error message to let the user try again or if wordpress only allows this one approach that kills everything that was written?

21. 21
Mung says:

With UD it takes you to a new error page and everything you wrote goes into the memory hole.

I have forgotten to fill in the captcha on numerous occasions. It sends me to a page asking me to go back and fill it in. The times I have done so, what I have written is still in the Comment box.

Other than that, I wonder if I’ve just never gotten one wrong, lol.

well, i just deliberately put in the wrong value and I was able to come back and finish my post.

Firefox 15.0.1

22. 22

Ah ha, so it is Firefox that saved you. Hmmmm. Maybe I’ll have to consider using that . . .

23. 23
es58 says:

Seqenenre @4 October 8, 2012 at 2:13 pm
Betting on a competitor God might displease the real God more than betting on no God at all.
Therefore the atheist choice might have some merit after all.

Might, but also, might not. Without more info, not sure how to assign odds to either side. So, if at best, we can say odds of might and might not are equal, then, added to the rest of the equation, it sounds like it should still carry the wager as Barry states?

24. 24
Bruce David says:

Eric:

Under your ‘logic’ a loving parent cannot punish a child for misbehaviors because that would mean that the parent doesn’t love the child. I don’t know if you have kids, but I can assure you this is most definitely false. Might want to rethink your position a bit . . .

Well first of all, no loving parent would ever condemn his or her child to an eternity of suffering, no matter how serious the crime committed. The idea the God would so condemn anyone for a finite amount of sin is in itself a contradiction to the idea of a loving God.

Secondly, there is a distinction between punishment and consequences. It is my belief (more than belief, actually—certainty comes closer to the mark) that unconditional love never judges and condemns (and therefore does not punish)*, but rather always supports. Setting consequences that arise naturally from action if done in love is a form of support.

*I am using the word “punish” in a somewhat narrower meaning than perhaps that of its common usage—punishment as I am using it here is a natural consequence of judgement and condemnation. It is retribution, or an evening of the score: “You did this, so you must suffer in return.” However, it is in this meaning that a Biblical God punishes us for our sins, so the usage is appropriate.

25. 25

Bruce David @24:

I understand your feeling with respect to some kind of eternity of suffering. I don’t believe in that kind of God either (for the most part). We’ll have to agree to disagree whether the Bible describes that kind of God. I don’t think it does. I think a loving parent-child relationship is a good place to start thinking about our relationship with God.

I also understand what you are saying about consequences that arise naturally. Two points come to mind:

We almost always set up arbitrary (I don’t mean capricious, just that they are not required by some natural law) consequences for actions. If you run a red light, what is the “natural consequence”? Well, someday you might cause a terrible accident. But our society doesn’t think that is good enough of a deterrent, so we set up an arbitrary imposed “punishment” of a ticket. If my kid refuses to do his homework, what is the “natural consequence”? Well, he won’t learn and may not reach his potential, or get into a good university, etc. But that kind of long-term natural consequence isn’t particularly motivational or helpful, so I impose a more near-term arbitrary “punishment,” namely that he can’t play on his xBox all evening if he doesn’t get his homework done. He thinks this is terrible punishment and that we are unfair and awful parents. Yet we, knowing more than he, understand that ultimately it will work to his benefit; and we of course still love him, notwithstanding the imposed punishment.

So the first point is that imposed consequences — ones that are not necessarily the “natural” consequences of an action — are extremely common, even necessary, in our society and our families. We do it all the time. We need to do it all the time for the normal functioning of society. It certainly does not mean that we don’t care or don’t love the individual who experiences the punishment.

Second point: Completely on the opposite side of the spectrum, why would we think that some of the harsher punishments (torment after this life, for example) that are supposedly dished out by God are imposed punishments? Seems to me perfectly possible that some of the long-term punishments described by God are exactly as you think they should be — natural consequences of one’s actions. If I spend my life in rebellion to God’s will, might the end result when I get to the next life and realize what I have done be that I am ever after tormented spiritually and mentally with the agonizing thoughts of what I have done? Perhaps God is not threatening us with a punishment he is going to impose, but rather helpfully describing for us what the natural consequence of certain actions will be.