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CHAPTER 23: Probability
According to the doctrine of chance, you ought to put yourself to the trouble of searching for
the truth; for if you die without worshipping the True Cause, you are lost.109
Chance and the sovereignty of God? Gambling and apologetics? Do these things really belong in
the same course? Is the study of chance appropriate at a Christian university?
I suppose a complete answer to such questions could take a long time. Here are a few of my brief
responses for you to ponder.
I believe in the doctrine of the sovereignty of God: God is in control of the events of our lives, and
knows our future. But how He does that remains rather mysterious to me, so from my human perspective,
"chance" is a description of how some things appear to happen. Consider Proverbs 16:33, "The lot is cast
into the lap, but its every decision is from the LORD."
Gambling is probably an unwise thing to do for an individual player: you will almost certainly lose
in the long run. Studying some probability helps you understand why the casino in Las Vegas is a
successful business. The casino isn't really gambling in the sense that they might lose; probability says that
they are essentially guaranteed to make big profits. By the way, the same thing is basically true about
insurance companies. If they play the odds right concerning your health, driving record, etc., they'll make
good profits.
Speaking of your health, the next time your doctor prescribes medicine for you, ask her if she is
absolutely certain (remember Descartes) that it will work. Not 99% sure, but absolutely certain. In fact,
maybe you should back up and ask if she is absolutely certain about the diagnosis that led to the
prescription. The point is, while we might prefer to be certain, we often act on the basis of probability.
Pascal and his friend Fermat are often considered the fathers of probability theory. While others
had considered a reasoned approach to gambling before them, it was Pascal who made a successful attempt
at developing the rudiments of a mathematical theory like Euclid had done for geometry. His goal was "to
reduce to an exact art, with the rigor of mathematical demonstration, the incertitude of chance, thus creating
a new science which could justly claim the stupefying title: the mathematics of chance."110 And he used his
new-found theory in his apologetic for the Christian faith.
Our first approach to probability will consider situations in which there are a number of possible
outcomes, each of which is just as likely to occur as any other. One such example is rolling a die ("die" is
singular; "dice" is plural.) There are six equally-likely outcomes possible. As another example, consider a
box containing 4 red pencils, 5 green pencils, 3 blue pencils, and 1 tan pencil. The pencils are identical
except for color. Close your eyes and pick a pencil from the box. You are just as likely to pick one pencil as
any one of the other pencils.
If a situation has n equally likely outcomes, and m of them are favorable to the happening of a
m
certain event, then the probability of that event is n . According to this definition, it is clear that
probabilities are numbers from 0 to 1. If the probability of an event were 0, there would be no favorable
outcomes, i.e., the event could never happen. For instance, try rolling a "10" with one die. Or try picking a
yellow pencil out of the box. At the other extreme, an event with a probability of 1 is certain to occur since
every outcome is favorable.
1
Most events have probabilities between 0 and 1. The probability of rolling a "3" is 6 . The
3
probability of rolling an even number is 6 since 2, 4, and 6 are even number rolls. The probability of
3
picking a blue pencil out of the box is
since there are 3 blue pencils, and 13 pencils total.
13
109Pascal,
110Pascal,
Pensees, p. 87.
quoted in Kline, Mathematics for the Non-mathematician, p, 522.
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Sometimes thinking about equally-likely outcomes can require extra care. Suppose you toss two
coins. There are three possible outcomes: two heads, two tails, and one of each. Are these three outcomes
equally likely? Stop reading and try this 20 or 30 times. Does each outcome occur with about the same
frequency? To think this situation through more carefully, suppose one coin was shiny and the other was
dirty [we just need a way to tell them apart]. Now the possible outcomes are: two heads, two tails, shiny
heads and dirty tails, dirty heads and shiny tails. These are the four equally likely outcomes. So, for
1
instance, the probability of getting two heads is 4 because there is one way to get two heads out of the
2
four equally likely possible outcomes. Similarly, the probability of getting one head and one tail is 4
because there are two ways to get a head and a tail out of the four equally likely outcomes possible. So
two heads, two tails and one of each are not equally likely outcomes. Since this is a theoretical result, it may
not match exactly with your experience, but your experience would typically be close. Yes, you are right -that's another probability statement about a probability.
Independent events
Thinking about tossing two coins leads to another important probability concept: independence.
Two events are independent if the outcome of one of them does not influence the outcome of the other. If
A and B represent two events, the official definition of A and B being independent is that the probability of
A and B both occurring is the probability that A occurs times the probability that B occurs. The
mathematical notation to say this is:
P (A and B) = P (A) • P (B).
This is exactly what happens with the two coins above. If A = the shiny coin comes up heads and B
= the dirty coin comes up heads, then the probability of getting two heads is
1
1
1
P (A) • P (B) = 2 • 2 = 4 .
Example 1: Find the probability of rolling two dice and getting two 5’s.
Solution: The rolls are independent. The 6 possibilities when you roll a die are equally likely, so
1
the probability of rolling a 5 on any roll is 6 .
1 1
1
So the probability of rolling two 5’s is 6 • 6 = 36 .
Example 2: Find the probability of rolling a die and getting a 2, and then tossing a coin and getting
a tails.
Solution:
1 1
1
6 • 2 = 12
.
If A and B are not independent events, you can still find the probability that they both occur.
In this more general case, however, you assume that one of the events has occurred. In this case, the rule
appears this way:
P (A and B) = P (A) • P (B, assuming A has occurred).
For instance, suppose we return to the box discussed earlier containing 4 red pencils, 5 green pencils, 3
blue pencils, and 1 tan pencil. Suppose you select 2 pencils from the box, one after another. [Do not put
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the first pencil back in the box.] What is the probability that the first pencil you select is red, and the
second pencil is green?
The probability that the first pencil is red is 4/13 since 4 of the 13 pencils in the box are red.
However, when you go to select the second pencil, there are only 12 pencils in the box. So, selecting a
green pencil after having selected a red pencil first has a probability of 5/12. So the probability of
selecting a red pencil first and then a green pencil is 4/13 • 5/12 = 20/156.
Example 3: Find the probability of selecting 2 pencils from the box above, the first pencil being
blue and the second being red.
Solution: P (blue and red)
=
P (blue) • P (red, assuming a blue pencil was picked first)
=
3/13 • 4/12 = 1/13.
Here's a very different example from the pen of C. S. Lewis. In an article entitled, "Modern Theology and
Biblical Criticism", Lewis discusses his views on some attempts in Biblical criticism to develop a theory of
how a certain passage in the Bible came to be written. The particular example Lewis gives is not
important for our purposes. Here are his words about the probability involved:
"When a critic reconstructs the genesis of a text he usually has to use what may be
called linked hypotheses. . . . Now let us suppose . . . that the first hypothesis has
probability of 90 per cent. But the two together don't still have 90 percent, for the
second comes in only on the assumption of the first. You have not A plus B; you
have a complex AB. And the mathematicians tell me that AB has only an 81 per cent
probability [90% times 90% = .90 x .90 = .81 = 81%-ed.] I'm not good enough at
arithmetic to work it out, but you see that if, in a complex reconstruction, you go on
thus superinducing hypothesis on hypothesis, you will in the end get a complex in
which, though each hypothesis by itself has in a sense a high probability, the whole
has almost none.111
Believe it or not, the probability theory inspired by the study of tossing coins and rolling dice turns
out to have very useful results. Here are two examples about which you are likely to hear on the TV news.
The first is political polls. As an election approaches, surveys of a relatively small number of voters are
used to infer who is currently supported by a majority of the voters. Probability theory allows us to
compute how reliable the inferences are.
As a second example, consider the testing of a new treatment for a disease. By counting how many
people receiving the new treatment are cured, an inference can be made about the probability that the new
treatment is an improvement over the old treatment.
The idea of probability has also been used in apologetics. For instance, Josh McDowell in Evidence
That Demands a Verdict discusses the probability that Messianic prophecies of the Old Testament would be
fulfilled in a single person. Quoting another author, he concludes
Now these prophecies were either given by inspiration of God or the prophets just wrote
them as they thought they should be. In such a case the prophets had just one chance in
1017 of having them come true in any man, but they all came true in Christ.
111
C.S.Lewis, The Seeing Eye and Other Selected Essays for Christian Reflections, pp 218-219
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This means that the fulfillment of these eight prophecies alone proves that God inspired the
writings of those prophecies to a definiteness which lacks only one chance in 1017 of being
absolute.112
For a philosophical discussion of this use of probability, see J. P. Moreland, Scaling the Secular City, pp. 5975. Moreland also briefly discusses Pascal's Wager (pp. 131-132), and it is with an analysis of this argument
that we will complete our study of probability.
Pascal's Wager
First, we need one more concept. It is called "expected value". The terminology might be
misleading if you think about the way those words are used in ordinary speech. The technical term is, for
our purposes, almost synonymous with the mean which we defined in the previous section. The expected
value in a situation is a weighted average, the weights being the probabilities of the various outcomes. For
instance, when you roll a die, there are 6 possible outcomes, and they are equally likely. In this case, the
expected value would be found by computing:
1
1
1
1
1
21
1
(1 • 6 ) + ( 2 • 6 ) + (3• 6 ) + (4• 6 ) + (5 • 6 ) + (6 • 6 ) = 6
= 3.5.
That is, 3.5 is the "average" result when you roll a die. Of course, you can't actually roll a 3.5, but
that's OK; it's an average. So "expected value" does not mean " a value I expect to get".
In it's original setting, gambling, expected value had to do with what you could expect to win in a
game. For instance, suppose I agree to pay you $1 for every dot that's on the top of the die when you roll it,
i.e., if you roll a 5, you get $5. However, to play the game, you must pay me $4. On the average, you would
expect to win $3.50 per time you played the game, but after subtracting the $4 it cost to play, you would net,
on the average, -$.50. That is, I could expect to gain $.50 on the average, every time I could talk you into
playing this game. Granted, you might "get lucky" once in a while, and come out ahead, but in the long
run, probability is in my favor. That's why there is very little "gambling" involved in the operation of a
casino: the players are gambling, but the casino is almost certain to win big in the long run.
Another way to calculate this expected value would be to figure in the cost of the game at the start.
This way, you would think of rolling a 5 as actually getting you only $1: -$4 to play, and then winning $5, a
net of -$4 + $5 = $1. Rolling a 1 would mean you would actually lose $3. Multiplying the net by its
probability would look like this:
1
(-$3 • 6
1
1
1
1
1
-$3
) + ( -$2 • 6 ) + (-$1• 6 ) + ($0• 6 ) + ($1 • 6 ) + ($2 • 6 ) = 6
= -$.50.
Example 4: A box contains 3 red pencils, 2 green pencils and 10 orange pencils. Consider the
following game: without looking, you pick a pencil from the box. If the pencil you pick is
green, you get $3; if it is red, you get $2. It costs $1 to play this game. Find the expected value.
Solution: expected value
112Peter
= expected winnings -- cost
2
3
= $3 • 15 + $2 • 15 – $1
= – $0.20
or
Stoner, quoted in Josh McDowell, Evidence That Demands a Verdict, Here’s Life Publishers, San
Bernardino, CA, 1979, p. 167.
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Chapter 23
expected value
=
=
2
3
10
$2 • 15 + $1 • 15 + (–$1) • 15
– $0.20
The concept of expected value is useful in other settings as well. Someone might call these
situations "non-gambling" examples, but I'll let you be the judge of that.
Consider the situation of a real estate agent. This person agrees to try to sell a house for people
wanting to sell it. The real estate agent spends money (and time) working to sell the house, but the owners
are only required to pay for these services if, in fact, the house sells. It is typical that the real estate agent
gets 6% of the selling price if she sells it by herself, and 3% of the selling price if another agent is involved in
the sale (representing the buyer). Suppose a certain real estate agent knows from past experience that she
has a 30% chance of selling any house by herself, a 25% chance of selling the house with another agent
involved, and a 45% chance of not selling the house. What is the expected value or income if she agrees to
try to sell a house for $350,000?
outcome
income
probability
sold by herself
sold with another
not sold
$21,000
$10,500
$0
.30
.25
.45
expected value = ($21,000) (.30) + ($10,500) (.25) + ($0) (.45)
= $6,300 + $2,625 + $0
= $8,925
What's the significance of an expected value of $8,925? This is the average amount of income that
the agent can "expect" to get from $350,000 houses. This is helpful to know because it suggests something
about how much money the agent would be willing to invest in advertising a house. In some cases (when
the house sells), such advertising costs will be easily covered. But if the house does not sell, they are a
complete loss. Expected value helps to balance out such occurrences over time. [By the way, this income
doesn't typically go directly to the agent. A portion goes to her agency for overhead.]
So how did Pascal use this concept of expected value in apologetics? The relevant passage from the
Pensees is usually referred to as Pascal's Wager. Students of Pascal have come up with a variety of ways of
interpreting and developing Pascal's Wager. For a collection of scholarly essays on the Wager, see
Gambling on God, Jeff Jordan, ed., Lanham, MD: Rowman & Littlefield Publ., 1994. Pascal presents many
arguments in the Pensees to give evidence to the claim that the God of the Bible exists. But for the sake of
this argument he assumes that "God exists" and "God does not exist" are equally likely events. That is, in
colloquial terms, it is "a toss up". And each of us must bet on the outcome of the toss, i.e., we must chose
between the two options in life (that is, we either live as if God exists or as if He doesn't). Unlike a game,
however, we cannot choose not to play "Life".
In one of the essays in Gambling on God, Thomas Morris writes:
In his attempt to recommend a Christian world view to his unbelieving
contemporaries, Blaise Pascal hit upon a rare form of argument, an
argument that I believe was meant to move them into a better position to
have the sort of experience of the reality of God they lacked and of the truth
of the Christian message about right relations with God. . . . Pascal devised
an argument to show us that we all ought to bet our lives on God.113
113
Thomas V. Morris, "Wagering and the Evidence", Gambling on God, p.48.
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Chapter 23
In Pascal's words, "Let us weigh the gain and the loss in wagering that God is."114 Recalling that
the mean or expected value is like a weighted average, Pascal is essentially going to compute the expected
value of this "game" when a person bets about God's existence. The gain and loss in this case is not in terms
of money, as in our previous examples. Pascal attempts to measure the gain and loss in "happiness". In this
analogy, your bet (what you pay to play the game) is how you live your life when you believe that God
exists. This is what you will "lose" if God does not exist (the coin comes up tails). Pascal suggests that the
amount of "happiness" you would lose is finite at worst. He puts it this way: "Now, what harm will befall
you in taking this side? You will be faithful, honest, humble, grateful, generous, a sincere friend, truthful.
Certainly you will not have those poisonous pleasures, glory and luxury; but will you not have others?"115
If it turns out that God exists (the "coin" comes up heads, just like you called it), you win "an eternity of life
and happiness". The expected value, then, looks something like this:116
Bet
Probability
Payoff
Cost
Expected Value
Christianity
.5
Infinite
Finite
Infinite
Atheism
.5
Finite
Finite
Finite
Since no matter what finite amount you subtract from infinity you get infinity, Pascal concludes
that the expected value is positive infinity. The argument has an obvious conclusion: "If you gain (win),
you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is."117 "I will tell you
that you will thereby gain in this life, and that, at every step you take on this road, you will see so great
certainty of gain, so much nothingness in what you risk , that you will at last recognize that you have
wagered for something certain and infinite, for which you have given nothing."118
Pascal concludes this passage with the following statement:
If my words please you and seem cogent, you must know that they come from a man who
went down upon his knees before and after to pray this infinite and indivisible being [God],
to whom he submits his own [being], that he might bring your being also to submit to him
for your own good and his glory..."119
114Pascal,
Pensees, p. 84.
Pensees, p. 86.
116 Thomas V. Morris, Gambling on God, p. 55
117Pascal, Pensees, p. 84.
118Pascal, Pensees, p. 86.
119Pascal, Pensees, p. 86.
115Pascal,
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CHAPTER 23: Probability
181
Homework
1. Suppose you roll a die. What is the probability you roll a 4?
2. Suppose you roll a die, then roll it again. What is the probability that the first roll is a 3, and the second
roll is also a 3?
3. A box contains 3 red pencils, 5 blue pencils, and 7 green pencils. You pick one pencil from the box. Find
the probability that it is:
a.
b.
c.
d.
blue
green
red or green
not red
4. A college student body consists of 1300 Caucasians, 300 Asian, 200 Afro-Americans and 300 Hispanics.
If a student is selected arbitrarily, what is the probability that the student is:
a. Afro-American
b. Asian or Caucasian
c. not Caucasian
d. Hispanic
5. Find the expected value of the number of heads you get when you toss two coins.
6. Consider a game in which you toss 2 coins. You win $1 if you get 1 head and $3 if you get 2 heads. Find
the expected value of this game.
7. Suppose it costs $2.25 to play the following game: you toss two coins and are paid $2 for each "head".
What is the expected value for this game?
8. Consider a game in which you roll a die. You win $1 if you roll a 4, $2 if you roll a 5, and $4 if you roll a
6. What is the expected value for this game?
9. A box contains 1 red pencil, 2 green pencils, and 7 blue pencils. Consider the following game: without
looking, pick a pencil from the box; if it is red, you win $5, and if it is green, you win $2. Suppose it
costs only $1 to play this game. What is its expected value?
10. A box contains 3 red pencils, 4 green pencils, and 13 blue pencils. Consider the following game:
without looking, pick a pencil from the box; if it is red, you win $12, and if it is green, you win $3.
Suppose it costs only $2 to play this game. What is its expected value?
11. Suppose drilling an oil well yields oil 2% of the time worth $2,500,000 , yields natural gas 5% of the time
worth $325,000 and is dry (worth nothing) 93% of the time. Suppose it costs $25,000 to drill an oil
well. Find the expected value of a well.
12. Suppose drilling an oil well yields oil 4% of the time worth $1,500,000 , yields natural gas 9% of the time
worth $200,000 and is dry (worth nothing) 87% of the time. Suppose it costs $100,000 to drill an oil
well. Find the expected value of a well.
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Chapter 23
Selected Answers
1
1.
6
1
2.
36
7
10
12
5
b. 15
c. 15
d. 15
3.
a. 15
2
16
8
3
4.
a. 21
b. 21 c. 21
d. 21
5.
1
6.
$1.25
7.
-$.25
9.
-$.10
11.
$41,250
182