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The lecture will begin shortly.
Exam 3 preliminary results
None of the 33 items was clearly ineffective.
The exam was more difficult than anticipated.
• 4 items were rated “difficult”
• 28 were “moderately difficult”
• 1 was “easy”
An adjustment will be made to the scores.
Lecture 29
Today’s lecture will cover material related to
Sections 16.6 and 18.5
1. Review expected value (Section 16.6)
2. Expectation and personal decisions (Section 18.5)
1. Review of Expected Value
Probability distribution
A probability distribution is a list of
• all possible outcomes, and
• the probabilities of those outcomes
Example
A couple has two children. Find the probability
distribution for the number of boys.
• # of boys = 0 with probability ¼ (GG)
• # of boys = 1 with probability ½ (GB or BG)
• # of boys = 2 with probability ¼ (BB)
Expected value
If you know the probability distribution, you can find
the expected value (or “expectation”):
• multiply each outcome by its probability, and
• add them up
Example
A couple has two children. Find the expected number
of boys.
0×¼=0
1×½=½
The expected value is 0 + ½ + ½ = 1.
2×¼=½
Interpretation of Expected Value
The expected value is the average value of the outcome
over the long-run, if the experiment were repeated many
times.
For the last example, suppose that took a very large,
random sample of families with two children.
And suppose that we recorded the number of boys in
each of these families:
0, 2, 1, 0, 1, 1, 2, 1, 1, 0, 0, 2, 1, …
The average of all of these numbers would be the
expected value, which in this case is 1.
Another example
Suppose you roll a single die. What is the expected value?
outcome
prob.
1
2
3
4
5
6
1/6
1/6
1/6
1/6
1/6
1/6
outcome × prob.
1/6
2/6
3/6
4/6
5/6
6/6
Expected value = 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6
= 21/6 = 3.5
Note that this is not one of the possible outcomes. It’s the
average value of the outcomes if the experiment is repeated
many times.
Weird example
“St. Petersburg paradox”
A gentleman from Russia approaches you and says:
“I will flip this coin until it comes up ‘heads.’
If I get heads on the first try, I will give you
$2. If it takes two tries, I will give you $4. If
it takes three tries, I will give you $8, and so
on.” (For each additional try, the
prize is doubled.)
How much would you pay to play this game?
$5
$10
$25
$50
List the possible outcomes and their probabilities:
Outcome ($)
H
HH
HHH
HHHH
HHHHH
2
4
8
16
32
prob.
outcome × prob.
1/2
1/4
1/8
1/16
1/32
1
1
1
1
1
…and so on
Expected value = 1 + 1 + 1 + 1 + 1 + 1 …
=∞
Your expected winnings are infinite.
So why were you unwilling to pay lots of money to play
this game?
2. Expectation and personal decisions
It makes sense to base financial decisions purely on
expectation or expected value if the experiment is truly
going to be repeated many times.
For example, your average gains (or losses) will be
approximately equal to the expected value if you
• play the lottery every week
• are a professional gambler
• sell insurance to many customers
• operate a gambling casino
Personal decisions
Individuals who are making one-time decisions typically
do not make those decisions purely on the basis of
expected values.
When making personal decisions, people look at both
the outcome and the probability, not just the
outcome × probability.
For example, suppose you could buy a raffle ticket for
$10 that would give you a 1 in 3,000 chance to win a
new car worth $25,000. Would you buy it?
Your expected winnings in this raffle are
$ 25,000 × 1/3000 = $ 8.33
From a standpoint of expectation, it does not make sense to
buy the raffle ticket, because $8.33 is less than $10.
But many people would buy the raffle ticket.
Many are willing to trade a small amount of money for a
small chance to win a large amount of money.
It’s because losing a small amount of money makes little
difference in their lives, whereas winning a large amount
would make a substantial difference.
Another example
Suppose I offer you two choices:
• a check for $25,000 or
• a 1 in 40 chance to win $1,000,000
Unless you are already quite wealthy, you would probably
take the check for $25,000.
In each case, however, the expected value of the prize is
the same:
$ 25,000 × 1 = $ 25,000
$ 1,000,000 × 1/40 = $ 25,000
If individuals made personal decisions based on expected
values, they would be indifferent to these two choices.
But they are not indifferent; most would take the check.
Health insurance
Suppose there is a 1 in 50 or 2% chance that you will have
a serious illness or injury this year requiring a hospital stay
and medical care that costs $50,000 or more.
Would you pay $100 per month for a medical plan that would
cover the cost of medical care in those extreme circumstances?
Your expected cost for the year if you buy insurance is
$ 1,200 × 1 = $ 1,200
Your expected cost for the year if you don’t buy insurance is
$ 50,000 × 1/50 = $ 1,000
Health insurance is always more expensive than the
expected medical bill. But many people would buy it
anyway, because it gives them peace of mind.
When expected values matter
Who would be willing to accept $1,200 and then agree to
pay all your hospital bills next year, if you should have a
serious illness?
An insurance company.
The company has a large number of customers.
These customers function as repeated experiments.
The average hospital bill that they will have to pay per
customer will be close to the expected value.
So it is profitable for them to assume the risk for large
numbers of people, if the expected value of their customers’
claims is less than the selling price of their coverage.
Why do people gamble?
Gambling is a losing proposition for the gambler.
In pure games of chance (slot machines, lotteries, roulette,
etc.) the expected values always favor the house.
So why do people gamble?
A decision to gamble can be rational if
• the gambler derives some non-monetary benefit
(e.g. fun or excitement), and
• the gambler does not gamble too often, given
his or her financial situation
Problem gambling
If a person gambles too often, the law of expected values
inevitably takes over, causing the gambler to steadily lose
money at a greater rate than he can afford.
The problem gambler makes a small decision to gamble
“just this once.” But he does it over and over.
Each decision could be rational if it were an isolated event.
But taken together, these individual decisions can bring
disastrous loss.
A problem gambler fails to place these momentary decisions
into the broader context of life.