Home > Science and Faith, World Views > Foundations: 5) Pascal’s Wager

Foundations: 5) Pascal’s Wager

Pascal’s Wager has been controversial since it was posed over 300 years ago.  It has been widely analyzed, misunderstood, and misapplied to “prove” the existence of God.  It does no such thing.  Rather,

“Pascal’s Wager is an attempt to justify belief in God not with an appeal to evidence for his existence but rather with an appeal to self-interest. It is in our interests to believe in the God of Christianity, the argument suggests, and it is therefore rational for us to do so.”  (http://www.philosophyofreligion.info/theistic-proofs/pascals-wager/ )

Before proceeding further, it is important to recognize that it is impossible to believe in the “God of Christianity” without believing in the foundational tenets of Christianity.  Any contradictory belief is non-Christian.  For example, Jews and Christians believe in the same God, but Jews do not believe in the foundational Christian teaching that Christ Jesus is the incarnate Son of God.  Therefore the two belief systems are not identical and must be treated as different for purposes of this discussion.  For that reason, I will replace “God of Christianity” with the more comprehensive term “Christianity” for the remainder of this discussion.  This substitution does not affect Pascal’s logic, which goes as follows (paraphrased):

(1) Christianity is either true or not.

(2) If Christianity is true and one believes, then that person receives an infinitely great reward. (See Case 1 in following Table)
If Christianity is false, then one loses little or nothing whether one believes or not.  (See Cases 2 and 4 in following Table)

(3) If Christianity is true and one does not believe, then that person receives an infinitely great punishment.  (See Case 3 in following Table)
If Christianity is false, then one loses little or nothing whether one believes or not.  (See Cases 2 and 4 in following Table)

(4) It is better to either receive an infinitely great reward or lose little or nothing than it is to either receive an infinitely great punishment or gain little or nothing.


(5) It is better to believe in Christianity than it is not to believe in Christianity.

(6) If one course of action is better than another then it is rational to follow that course of action and irrational to follow the other.


(7) It is rational to believe in Christianity and irrational not to believe in Christianity.

The following table diagrams the above wager in terms of decision theory:

Pascal's Decision Matrix

Assumptions:  Christianity must be either true or not true.  (See Step 1 of Pascal’s Wager)

Probability that Christianity is true:  I picked a small number to illustrate a point.  I assumed there are 5 competing major religious beliefs in the world, each equally probable.

Probability that Christianity is not true:  1 – (Probability that Christianity is true)

Benefit of Heaven:  This number should really be infinity, but that would not allow the math to be demonstrated.  So I picked the smallest number I could think of as follows:  According to scientists the age of the Universe is roughly 14 billion years, so I equated that with eternity, allocating 1 point per year.  Eternity in Heaven is actually much, much longer and worth many, many times that!

Trials on earth:  I allocated -1 point per year of life, assuming an 80 year life-span.

Net:  Benefit value – Cost value

Expected value:  (Net value) x (Probability that Christianity is True); or not true as the case may be

The important number is the Net Expected Value.  (2,800,000,000 for Case (1), for example)  The higher the number, the more desirable the case.  The lower the number, the less desirable the case.  Clearly Case (1) is by far the most desirable, and Case (3) the least desirable.  What is Case (1)?  Believing in Christianity and finding it to be true.  What is Case (3)?  Not believing in Christianity and finding it to be true.  In comparison, if Christianity is not true it makes little difference whether we believe or not (Cases 2 and 4).  That is the heart of Pascal’s Wager.

Don’t like these numbers?  Fine.  Use your own estimates.  It will not change the end conclusion:  From a cost-benefit perspective, it is foolish not to believe in Christianity.

Careful examination of the table clearly illustrates the truth of Pascal’s Wager.  We can fiddle with the Assumptions all we want, but the benefits of Heaven and cost of Hell as described by Christianity will far outweigh any benefits or cost we may encounter in this life.


Many objections have been raised against Pascal’s argument, but none have refuted its fundamental truth.  Instead, all these objections have focused on theological details, and not the actual wager itself.  That is as one would expect:  Those who do not believe in Christianity will disagree with any conclusions reached on the basis of Christianity.  Some illustrative objections are examined below:



The existence of God is unlikely The probability for the existence of God is immaterial, because infinity – a very large number – times any probability, no matter how small, is always infinite.  Thus the expected value for Case (1) is always infinity, and for Case (3) it’s always negative infinity.
The probability for the existence of God is zero. This is an assumption that cannot be proven because we don’t know everything.  Therefore we cannot prove that something does not exist.In fact, there is significant evidence for the existence of God; therefore the probability for the existence of God must be greater than zero.
Criteria for admission to Heaven and Hell may not be distributed on the basis of belief.God may bestow different rewards on different people, perhaps based on different wagering strategies the individual employs.Different outcomes are not addressed, such as situational decision making that sometimes leads to “Godly” decisions and sometimes not. These positions are in conflict with fundamental Christian doctrine:  God offers only eternal salvation to believers and punishment for non-believers.  There are no other options, and we must choose.Salvation is a free gift of God, given only by His grace through our faith– not by any works we may perform.Therefore these are invalid objections and must be considered as part of Cases 2 and 4.
There are different utilities (costs and benefits associated with different choices) that are not addressed.The utility of Heaven and Hell are not infinite.If I wager something for God and He doesn’t exist, I’ve lost something. The utility of Heaven and Hell are not infinite in the Table, yet still overwhelm all other options.  This is not a significant issue.Eternal salvation or punishment is so much more valuable than any temporary cost or benefit.  Thus the temporary cost of any decision in this life, or the temporary benefit of the same, is negligible.Cases 1 and 3 clearly demonstrate this fact.
There may be many gods and the model doesn’t address them. Actually, the model does address them.  There is only one Christian God.  All other gods fall into the other column (Cases 2 and 4).Adjusting the probabilities between the two columns addresses this issue.  But the objection is immaterial because infinity times any small probability is still infinity, which is greater than any finite number.
We cannot choose our beliefs because we form beliefs on evidence.We may “fake” our beliefs to win the wager. These objections are contrary to basic Christian teaching.  God provides plenty of convincing evidence, but we choose not to accept the evidence.  God knows our hearts and cannot be fooled.  These situations must be considered as part of Cases 2 and 4.


Based on the exclusive and extraordinary claims of Christianity, it would be extremely foolish indeed to dismiss Christianity without further consideration.  The following short parable illustrates the danger:

Two friends were walking in a field.  One of them saw a bull, and said to the other, “I see a bull.  Let’s run away so he doesn’t kill us.”  But the second replied, “I don’t see a bull.  I don’t need to run.  Besides, what if the thing you’re seeing has stripes and isn’t a bull?”  The first man ran away; the second stayed and was gored to death by the bull.  It doesn’t matter whether you see the bull or not – he is still there.  It doesn’t matter whether you want to pretend that the bull you see is actually a harmless zebra – he is still a bull who will gore you to death.

All is not as it looks.

Other references


(http://en.wikipedia.org/wiki/Pascal%27s_Wager )


© 2012 notasitlooks

  1. March 26, 2012 at 4:37 pm

    I’m not sure if I agree with the interpretation of Pascal’s Wager as cited by the website. It has been a few years since I’ve read Pascal’s work, but I think that what he was getting at was that our emotions, not our intellect, is what is really holding people back from believing in God.

    The larger corpus of his work seems to give evidence to this. Pascal once published a paper that involved a glass tube, mercury, and a walk up a mountain to prove that vacuums existed. He thought that everyone would accept it, but there was so much backlash against it that he realized that humans are entirely intellectual animals -even when the clear-headed thinkers of the Enlightenment claimed that’s exactly what they were.

    • March 26, 2012 at 5:40 pm

      Thanks for reading and responding. I didn’t expect anyone to read it so quickly. It’s pretty heavy and I’m having trouble formatting one of the tables. The draft and the published versions are quite different.

      Please take a quick glance at the four references provided (three are at the end). The focus is the decision matrix and associated probabilities. They’re pretty consistent and representative of the corpus of commentary on Pascal’s Wager.


  2. April 7, 2012 at 10:03 pm

    The issue isn’t whether the decision matrix make sense or not. It does. The issue is that the website, as you quoted it, says:

    “Pascal’s Wager is an attempt to justify belief in God not with an appeal to evidence for his existence but rather with an appeal to self-interest.”

    I do not think that Pascal was trying to “justify” a belief in God. He was illustrating a point that he believed pretty strongly. Mainly, that the intellect isn’t what’s holding most people back from belief in God. The website you cited, I feel in misinterpreting Pascal.

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