Statistics and Data
Analysis
Professor William Greene
Stern School of Business
IOMS Department
Department of Economics
4-1/25
Part 4: Expected Value
Statistics and Data Analysis
Part 4 – Expected
Value
4-2/25
Part 4: Expected Value
Expected Value Expected Agenda
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4-3/25
Discrete distributions of payoffs
Mathematical expectation
 Expected return on a bet
 Fair and unfair games
Applications: Warranties and insurance
Litigation risk and probability trees
Part 4: Expected Value
The Expected Value
A random “experiment” has outcomes (payoffs) that
are quantitative, monetary for simplicity.
Probability distribution over M possible outcomes is
Probability
P1 P2 P3 … PM
Payoff
$1 $2 $3 … $M
Expected outcome (payoff) is
P1 $1 + P2 $2 + P3 $3 + … + PM $M.
Note: The “average” outcome = a weighted average
of the payoffs
4-4/25
Part 4: Expected Value
Fair Games
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4-5/25
Define: A game is defined to be a situation
of uncertain outcome with monetary payoffs.
Betting the entire company fortune on a new
product is a “game”
A fair game has Expected payoff = 0
“Fair” has no moral (equity) connotation. It
is a mathematical construction.
Part 4: Expected Value
A Fair ‘Game’ Has Zero Expected Value
I bet $1 on a (fair) coin toss.
Heads, I get my $1 back + $1.
Tails, I lose the $1.
Expected value = Σi=payoffs Pi $i
E[payoff]=(+$1)(1/2) + (-$1)(1/2) = 0.
This is a “fair” game.
4-6/25
Part 4: Expected Value
A Risky Business Venture
4 Alternative “Projects:” Success depends on economic
conditions, which cannot be forecasted perfectly.
For each project,
Expected Value = .9*(Result|Boom) + .1*(Result|Recession)
(Probability)
Beer
Fine Wine
Both
T-bill
4-7/25
Boom
(90%)
-10,000
+20,000
+10,000
+3,000
Recession Expected
(10%)
Value
+12,000
-7,800
-8,000
+17,200
+4,000
+9,400
+3,000
+3,000
Part 4: Expected Value
Which Venture to Undertake?
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4-8/25
Assume the manager cares only
about expected values (i.e., is
indifferent to risk)
BOTH surely dominates T-bill.
 T-bill produces a certain
+3,000
 BOTH > T-bill in either state.
BEER has a negative expected
value. Dominated by the other 3.
Choice is between FINE WINE and
BOTH. Based only on expectation,
choose FINE WINE
Why might she not choose FINE
WINE? It’s more risky. It might
lose a whole lot of money. The
initial assumption is unrealistic.
People do care about ‘risk,’ i.e.,
variation in outcomes.
Part 4: Expected Value
American Roulette
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Bet $1 on any of the 38 numbers.
If it comes up, win $35. If not, lose the $1
E[Win] = (-$1)(37/38) + (+$35)(1/38)
= -5.3 cents.
Different combinations (all red, all odd, etc.) all
return -$.053 per $1 bet.
Stay long enough and the wheel will always
take it all. (It will grind you down.)
(A twist. Why not bet $1,000,000. Why do
casinos have “table limits?”)
18 Red numbers
18 Black numbers
2 Green numbers (0,00)
4-9/25
Part 4: Expected Value
The Gambler’s Odds in Roulette
Every bet loses 5.3 cents/$
4-10/25
Part 4: Expected Value
The Gambler’s Ruin
4-11/25
http://en.wikipedia.org/wiki/Roulette
Part 4: Expected Value
Caribbean Stud Poker
4-12/25
Part 4: Expected Value
The House Edge is 5.22%
These are the returns to the player.
It’s not that bad. It’s closer to 2.5% based on a simple betting strategy.
4-13/25
http://wizardofodds.com/caribbeanstud
Part 4: Expected Value
The Business of Gambling
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Casinos run millions of “experiments” every day.
Payoffs and probabilities are unknown (except on slot
machines and roulette wheels) because players bet
“strategically” and there are many types of games to
choose from.
The aggregation of the millions of bets of all these
types is almost perfectly predictable. The expected
payoff to an entire casino is known with virtual
certainty.
The uncertainty in the casino business relates to how
many people come to the site.
http://www.foxnews.com/us/2014/09/09/decades-standing-pat-made-atlanticcity-loser-say-veteran-workers/
4-14/25
Part 4: Expected Value
Triple Damages in Antitrust Cases
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4-15/25
Benefit to collusion or other antisocial activity is B
Probability of being caught is P
Net benefit:
 If they just have to give back the profits:
B-P*B = (1-P)*B which is always positive!
 Under the treble damages rule:
B–3*P*B = (1-3P)*B might still be > 0 if P < 1/3.
How to make sure the net benefit is negative: Prison!
Part 4: Expected Value
Actuarially Fair Insurance
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Insurance “policy”
 You pay premium = F
 If you collect on the policy, the payout = W
 Probability they pay you = P
Expected “profit” to them is
E[Profit] = F - P x W > 0 if F/W > P
When is insurance “fair?” E[Profit] = 0?
Applications
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4-16/25
Automobile deductible
Consumer product warranties
Health insurance
Part 4: Expected Value
A proposal to replace Fannie Mae
and Freddie Mac with government
created insurance for mortgage
backed securities funded by
actuarially fair insurance.
(Think FDIC.)
4-17/25
Part 4: Expected Value
4-18/25
Part 4: Expected Value
Since Hamman founded the Dallas-based company in 1986, SCA has grown into
the world's go-to insurer of stunts that give fans the opportunity to win piles of
dough if they make a hole-in-one, kick a field goal or sink a half-court shot. SCA
determines the odds for each contest and charges event sponsors a fee based
on the probability of someone succeeding, prize value and number of
contestants. If the contestants win, SCA pays them the full prize amount. If they
don't, SCA pockets the premium. "The primary distinction between us and
[traditional] insurance businesses is that they restore something economically
when something bad happens," Hamman says. "When you put on a promotion,
you're simply taking a position on the likelihood of an event occurring."
How do they ‘determine the odds?’
4-19/25
Part 4: Expected Value
Fitzgerald said he estimates that Hamilton's blast probably saved customers a
little more than $500,000. His company bought insurance for the promotion. The
company issued an explanation of the Grand Slam Payout, which said anyone
who purchased flooring or countertops starting Aug. 29 would get a refund if
Hamilton hit a bases-loaded home run during the promotion period, which was
to run until Sept. 28 or as soon as Hamilton hit a grand slam.
4-20/25
Part 4: Expected Value
Health Care Insurance Vocabulary
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Insurance companies provide insurance against adverse health
outcomes. Under new methodology, they also insure against future
adverse outcomes by insuring against costs of prevention.
Government guarantees that there will be a large pool of customers
and in return insurance companies agree to a rate structure.
Rates are ‘fair’ if the pool contains a good mix of ‘heavy’ users
(old, sick) and ‘light’ users (young, healthy) who will not use the
system (on average)
Adverse selection: Not enough young people join. The rates
become ‘unfair’ against the insurance companies. Insurers must
raise rates – then the situation becomes worse. Death spiral.
Moral hazard: People change their behavior because they have
insurance – and rates do not reflect the changed behavior. Again,
rates become unfair (negative profits).
4-21/25
Part 4: Expected Value
Rational Use of a Probability?
For all the criticism BP executives may
deserve, they are far from the only
people to struggle with such lowprobability, high-cost events. Nearly
everyone does. “These are precisely the
kinds of events that are hard for us as
humans to get our hands around and
react to rationally,”
Quotes from Spillonomics: Underestimating Risk
By DAVID LEONHARDT, New York Times Magazine,
Sunday, June 6, 2010, pp. 13-14.
4-22/25
E[benefit of building and operating rig]
= very low probability * very high cost
+very high probability * huge profits.
The expected benefit is positive under
any realistic calculations of costs and
profits. This insurance ‘market’ fails.
Food for thought: Why does it fail?
Part 4: Expected Value
Litigation Risk Analysis
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4-23/25
Form probability tree for decisions and outcomes
Determine conditional expected payoffs (gains or
losses)
Choose strategy to optimize expected value of payoff
function (minimize loss or maximize (net) gain.
Part 4: Expected Value
Litigation Risk Analysis: Using
Probabilities to Determine a Strategy
P(Upper path) = P(Causation|Liability,Document)P(Liability|Document)P(Document)
= P(Causation,Liability|Document)P(Document)
= P(Causation,Liability,Document)
= .7(.6)(.4)=.168. (Similarly for lower path, probability = .5(.3)(.6) = .09.)
Two paths to a favorable outcome. Probability =
(upper) .7(.6)(.4) + (lower) .5(.3)(.6) = .168 + .09 = .258.
How can I use this to decide whether to litigate or not?
Suppose the cost to litigate = $1,000,000 and a favorable outcome pays $3,000,000.
What should you do?
4-24/25
Part 4: Expected Value
Summary
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4-25/25
Expected value = average outcome (weighted by probabilities)
Expected value is an input to business decisions
“Games” can be fair or “unfair” (have negative expected value).
 Some agents worry about unfair games
 All casino games are unfair but people play them anyway.
 Product warranties are a hugely profitable unfair game.
Consumers do not know much about probabilities. (Or
about manufacturer warranties.)
Many decision situations involve certain costs and random
payoffs. The cost benefit test requires an evaluation of
expected values.
Decision makers also worry about risk (variance) and also
about the utility of payoffs rather than the payoffs themselves.
Part 4: Expected Value