# PHIL 101 Pascal's wager

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```					PHIL 101: Pascal’s wager

Dennis Earl (dearl@coastal.edu)
March 29, 2010

Who is Blaise Pascal?

Pascal (1623-1662)                     Pascal’s
calculator

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Pascal’s wager

Pascal’s thesis: It’s rational to believe that God exists.

Why you should accept it:
   Treat the issue of theism vs. atheism vs. agnosticism as a
gamble, or as a wager.
   It’s a good bet to be a theist.
   So it’s rational to be a theist.
   So there you go.

But why is it a good bet to bet on theism?

Our two choices (and we must take one or the other):
   Believe in God
   Don’t believe in God

The choices are evaluated on the basis of:
   the probability of being right with respect to each choice
   the payoff for being right
   the payoff for being wrong

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So there are four possible outcomes to consider:

Our choice is between believing G and not believing G, where G is
G       God exists, and rewards believers with an infinitely good payoff1
(something like eternal heaven), and punishes nonbelievers with
an infinitely bad payoff (eternal hell, say).

There are then four possible outcomes:
       believing G and G being true
       believing G and G being false
       not believing G and G being false
       not believing G and G being true
1 Note how this is slightly different than the presentation in the Kaufmann article--the payoff there was merely
“winning it all,” or “winning everything.” If you’re in the 11:30, Dr. Jones gave the argument somewhat
similarly. I’m doing it this way because it’s closer to Pascal’s own version, and it allows the for the argument
to work (Pascal thinks) even if probability of G is very, very low (as long as it’s not zero).

What are the payoffs for the four options?1

G is true                               G is false

Believe G                                payoff = ∞                              payoff = 5

Not believe G                            payoff = -∞                             payoff = 10

1 Here again, note the differences between this table of payoffs and what’s in the Kaufmann article (and in Dr.
Jones’ table, for those of you in the 11:30 section). For the payoffs in the right column, I chose those numbers
simply so that (1) they would be finite, but still positive (since you get something, I suppose, on those two
options), and (2) they would be different from each other (so the calculations on the following slides would be
easier to follow).

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So is it rational to believe G or not?

Which has the highest expected value (EV), believing G or
not believing G?

EV(for a bet) = [Pr(winning) × payoff for winning]
+ [Pr(losing) × payoff for losing]
– the cost of playing

[What’s the point here? Roughly, the expected value of a choice is the
good that one would expect to get by choosing that option. When we
don’t know what’s going to happen, the probabilities of what’s going to
happen need to be considered, and that’s what the formula does. Then,
whichever option has the highest expected value is the option that it
would be rational to choose.]

What are the relevant probabilities?

Pr(G is true):
Say it’s one in a million that G is true, or 1/1,000,000.1

Pr(G is false):
The probability that G is false is then (1-1/1,000,000).2

1 The probability here could be anything, Pascal says, as long as it isn’t zero. The chances of G being
true could be 1 in a trillion, but as will be seen, that won’t matter to Pascal’s argumentative move.

2 This should be easy to see, even if you haven’t learned this elsewhere. Probabilities range from zero (it
definitely won’t happen) to one (it definitely will happen). If you roll a 6-sided die, the chances of getting a
“2” is one in six, or 1/6. The chances of not getting a “2” is five in six, or 5/6, or (1-1/6).

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So now run the numbers (and let the cost of playing be 1).
The EV for believing G is:

EV(believing G) =
[Pr(G is true and believe G) × payoff(G and believe G)]
+ [Pr(G is false and believe G) × payoff(G is false and believe G)]
– cost of playing1

EV(believing G) =
= [(1/1,000,000) × ∞] + [(1-1/1,000,000) × 5] – 1
=∞+5–1
=∞
1 For sake of argument, let the cost of believing (i.e., making the bet) be 1. It doesn’t matter much, but
there presumably is some finite cost of believing. That goes into the calculation too.

The EV for not believing G is:

EV(not believing G) =
[Pr(G is false and don’t believe G) × payoff(G is false, and
don’t believe G)]
+ [Pr(G is true) × payoff(G is true and don’t believe G)]
– cost of playing

EV(not believing G) =
= [(1/1,000,000) × 10] + [(1-1/1,000,000) × -∞] – 1
= 0 + -∞ - 1
= -∞

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So…

The EV of believing G is infinitely good.
The EV of not believing G is infinitely bad.

So the EV of believing G is higher than not believing G.
So it’s rational to believe in God.

A simplified version of the payoffs where the argument still
favors rational belief in theism

G is true                                G is false

Believe G                             payoff = ∞                               payoff = something
finite

Not believe G                         payoff = something payoff = something
finite             finite

The point: As long as the payoffs for the options other than believing G and
G’s being true are finite (either finitely good or finitely bad), then the
expected value of believing G is still higher than not believing G.
See http://plato.stanford.edu/entries/pascal-wager/, section 4 for more on this. You can also do the
calculation yourself--just follow the formula on the previous slides, and the EV of believing G will still come
out higher than not believing G.

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Critical questions to consider (from the Kaufmann
article):

Question 1:

Can you really just decide or choose to believe something? Is
that a real belief that you wind up with?

Critical questions

Question 2:

What might God think about people who decide to believe
simply on the basis of “prudence,” or potential gain for
themselves?

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Question 3: The “many gods” objection
What about these possibilities?

   A god that rewards atheists and damns irrational believers
   A god that rewards agnostics and damns everyone else
   A god that rewards particular sorts of believers but damns
others
   Another kind of god entirely—a flying spaghetti monster,1
say—that rewards believers in it with an infinite payoff and
damns everyone else.
   A god that rewards the open-minded with an infinite payoff,
and damns the rest.
   The point: It looks rational to believe in each and every
possibility. But that in itself looks irrational.

1   See http://www.venganza.org/.

Summing up

 Pascal’s argument is a good example of calculating the
best decision based on uncertain probabilities and
outcomes.
 Does the argument work? These questions are crucial:
 Can you really make yourself believe something you
want to believe?
 What about the “many gods” objection? What are the
probabilities of those different possibilities relative to
each other?

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For next time:

The problem of evil

Reading: John Mackie, “Evil and omnipotence”

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