6.3 Probabilities with Large
Numbers
LEARNING GOAL
Understand the law of large numbers, use this law to
understand and calculate expected values, and
recognize how misunderstanding of the law of large
numbers leads to the gambler’s fallacy.
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6.3-1
The Law of Large Numbers
The Law of Large Numbers
The law of large numbers (or law of averages) applies to
a process for which the probability of an event A is P(A)
and the results of repeated trials do not depend on results
of earlier trials (they are independent).
It states: If the process is repeated through many trials,
the proportion of the trials in which event A occurs will be
close to the probability P(A). The larger the number of
trials, the closer the proportion should be to P(A).
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6.3-2
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6.3- 2
EXAMPLE 1 Roulette
A roulette wheel has 38 numbers: 18 black numbers, 18
red numbers, and the numbers 0 and 00 in green.
(Assume that all outcomes––the 38 numbers––have
equal probability.)
a. What is the probability of getting a red number on
any spin?
Solution:
a. The theoretical probability of getting a red number
on any spin is
number of ways red can occur
P(A) =
=
total number of outcomes
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18
= 0.474
38
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EXAMPLE 1 Roulette
A roulette wheel has 38 numbers: 18 black numbers, 18
red numbers, and the numbers 0 and 00 in green.
(Assume that all outcomes––the 38 numbers––have
equal probability.)
b. If patrons in a casino spin the wheel 100,000 times,
how many times should you expect a red number?
Solution:
b. The law of large numbers tells us that as the game is
played more and more times, the proportion of times
that a red number appears should get closer to 0.474.
In 100,000 tries, the wheel should come up red close
to 47.4% of the time, or about 47,400 times.
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Expected Value
Definition
The expected value of a variable is the weighted average
of all its possible events. Because it is an average, we
should expect to find the “expected value” only when there
are a large number of events, so that the law of large
numbers comes into play.
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6.3-5
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Calculating Expected Value
Consider two events, each with its own value and
probability. The expected value is
expected value =
(value of event 1) * (probability of event 1)
+ (value of event 2) * (probability of event 2)
This formula can be extended to any number of events by
including more terms in the sum.
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EXAMPLE 2 Lottery Expectations
Suppose that $1 lottery tickets have the following probabilities:
1 in 5 to win a free ticket (worth $1), 1 in 100 to win $5, 1 in
100,000 to win $1,000, and 1 in 10 million to win $1 million.
What is the expected value of a lottery ticket? Discuss the
implications. (Note: Winners do not get back the $1 they spend
on the ticket.)
Solution: The easiest way to proceed is to make a table (next
slide) of all the relevant events with their values and
probabilities. We are calculating the expected value of a
lottery ticket to you; thus, the ticket price has a negative value
because it costs you money, while the values of the winnings
are positive.
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6.3-7
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EXAMPLE 2 Lottery Expectations
Event
Ticket purchase
Value
Probability
-$1
Win free ticket
$1
Win $5
$5
Win $1000
Win $1 million
$1,000
$1,000,000
1
1
5
1
100
1
100, 000
1
10, 000, 000
Value × probability
($1)  1  $1.00
$1  15  $0.20
1  $0.05
$5  100
$1,000  1001, 000  $0.01
1
$1,000,000  10, 000, 000  $0.10
The expected value is the sum of all the products value ×
probability, which the final column of the table shows to be
–$0.64.
Thus, averaged over many tickets, you should expect to lose 64¢
for each lottery ticket that you buy. If you buy, say, 1,000 tickets,
you should expect to lose about 1,000 × $0.64 = $640.
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6.3-8
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TIME OUT TO THINK
Many states use lotteries to finance worthy causes such
as parks, recreation, and education. Lotteries also tend to
keep state taxes at lower levels. On the other hand,
research shows that lotteries are played by people with
low incomes. Do you think lotteries are good social
policy? Do you think lotteries are good economic policy?
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6.3-9
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The Gambler’s Fallacy
Definition
The gambler’s fallacy is the mistaken belief that a
streak of bad luck makes a person “due” for a streak
of good luck.
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EXAMPLE 3 Continued Losses
You are playing the coin-toss game in which you win $1 for
heads and lose $1 for tails. After 100 tosses, you are $10 in the
hole because you have 45 heads and 55 tails.
You continue playing until you’ve tossed the coin 1,000 times,
at which point you’ve gotten 480 heads and 520 tails.
Is this result consistent with what we expect from the law of
large numbers? Have you gained back any of your losses?
Explain.
Solution: The proportion of heads in your first 100 tosses was
45%. After 1,000 tosses, the proportion of heads has increased to
480 out of 1,000, or 48%.
Because the proportion of heads moved closer to 50%, the
results are consistent with what we expect from the law of large
numbers.
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6.3-11
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EXAMPLE 3 Continued Losses
Solution: (cont.)
However, you’ve now won $480 (for the 480 heads) and lost
$520 (for the 520 tails), for a net loss of $40.
Thus, your losses increased, despite the fact that the proportion
of heads grew closer to 50%.
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6.3-12
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Streaks
Another common misunderstanding that contributes to the
gambler’s fallacy involves expectations about streaks.
Suppose you toss a coin six times and see the outcome
HHHHHH (all heads). Then you toss it six more times and see
the outcome HTTHTH.
Most people would say that the latter outcome is “natural”
while the streak of all heads is surprising.
But, in fact, both outcomes are equally likely. The total
number of possible outcomes for six coins is 2 × 2 × 2 × 2 × 2
× 2 = 64, and every individual outcome has the same
probability of 1/64.
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Moreover, suppose you just tossed six heads and had to bet on
the outcome of the next toss. You might think that, given the
run of heads, a tail is “due” on the next toss.
But the probability of a head or a tail on the next toss is still
0.50; the coin has no memory of previous tosses.
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TIME OUT TO THINK
Is a family with six boys more or less likely to have a boy
for the next child? Is a basketball player who has hit 20
consecutive free throws more or less likely to hit her next
free throw? Is the weather on one day independent of the
weather on the next (as assumed in the next example)?
Explain.
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EXAMPLE 4 Planning for Rain
A farmer knows that at this time of year in his part of the
country, the probability of rain on a given day is 0.5. It hasn’t
rained in 10 days, and he needs to decide whether to start
irrigating. Is he justified in postponing irrigation because he is
due for a rainy day?
Solution: The 10-day dry spell is unexpected, and, like a
gambler, the farmer is having a “losing streak.” However, if we
assume that weather events are independent from one day to the
next, then it is a fallacy to expect that the probability of rain is
any more or less than 0.5.
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The End
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