Casinos don’t gamble.
Chapter 7C
The Law of Large Numbers
Expected Value
Gambler’s Fallacy
7-A
When tossing a fair coin, which of the
following events is most likely?
A. Getting 8 tails in 10 tosses
B. Getting 80 tails in 100 tosses
C. Getting 800 tails in 1000 tosses
The Law of Large Numbers
7-A
Applies to a process for which the probability of an
event A is P(A) and the results of repeated trials are
independent
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).
The Law of Large Numbers
This figure shows the results of a computer
simulation of rolling a die.
7-A
The Law of Large Numbers
A roulette wheel has 37 numbers: 18 are black, 18 are
red, and one green.
1. What is the probability of getting a red number
on any spin?
2. If patrons in a casino spin the wheel 100,000
times, how many times should you expect a red
number?
The Law of Large Numbers
7-A
In a large casino, the house wins on its blackjack games
with a probability of 50.7%. Which of the following events is
the most likely? Most unlikely?
A. You win at a single game.
B. You come out ahead after playing forty times.
C. You win at a single game given that you have
just won ten games in a row.
D. You win at a single game given that you have
just lost ten games in a row.
7-A
Expected Value
Expected Value
Consider two events, each with its own value and
probability.
Expected Value
= (event 1 value) • (event 1 probability)
+ (event 2 value) • (event 2 probability)
The formula can be extended to any number of
events by including more terms in the sum.
7-A
Expected Value
You have been asked to play a dice game.
You will roll one die and:
If you roll a 1, 2, 3 or 4, you will lose $30.
If you roll a 5, you will win $48
If you roll a 6, you will win $60
What is the expected value (to you) of this
game?
7-A
Expected Value
Consider the following game. The game
costs $1 to play and the payoffs are $5 for
red, $3 for blue, $2 for yellow, and nothing for
white. The following probabilities apply. What
are your expected winnings? Does the game
favor the player or the owner?
Outcome
Red
Blue
Yellow
White
Probability
0.2
0.4
0.16
0.78
7-A
Expected Value
7-A
You are given 5 to 1 odds against tossing three
heads with three coins.
This means that you win $5 if you succeed and
you lose $1 if you fail.
Find the expected value (to you) of the game.
NOTE: A game is said to be "fair" if the expected
value for winnings is 0, that is, in the long run, the
player can expect to win 0.
Expected Value
7-A
Example: Suppose an automobile insurance
company sells an insurance policy with an annual
premium of $200. Based on data from past
claims, the company has calculated the following
probabilities:
An average of 1 in 50 policyholders will file a claim of $2,000
An average of 1 in 20 policyholders will file a claim of $1,000
An average of 1 in 10 policyholders will file a claim of $500
Assuming that the policyholder could file any of the
claims above, what is the expected value to the
company for each policy sold?
Expected Value
Let the $200 premium be positive (income) with a
probability of 1 since there will be no policy without
receipt of the premium. The insurance claims will be
negatives (expenses).
The expected value is
(This suggests that if the company sold many,
many policies, on average, the return per policy is
a positive $60.)
7-A
Expected Value
7-A
An insurance company knows that the average
cost to build a home in a new subdivision is
$100,000, and that in any particular year, there is
a 1 in 50 chance of a wildfire destroying all
homes in the subdivision.
Based on these data, and assuming the
insurance company wants a positive expected
value when it sells policies, what is the minimum
that the company must charge for fire
insurance policies in this subdivision?
Expected Value
Textbook problems
Orange: pg448 #35, 38
Green: pg489 #45, 49
7-A
Homework Chapter 7C
Orange: Pg 445
Quick Quiz: 1, 2, 4, 5, 6
Exercises: 1, 2, 9, 15, 19, 31, 35, 38
Green: Pg 485
1, 2, 4, 5, 6, 11, 12, 19, 25, 29, 41, 45, 49
In a family of six children, which sequence do
you think is more likely?
a. MFFMFM
b. MMMMMM
c. Both are equally likely