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Expected Value
Section 3.5
Definition
• Let’s say that a game gives payoffs
a1, a2,…, an with probabilities
p1, p2,… pn. The expected value ( or
expectation) E of this game is
E = a1p1 + a2p2 + … + anpn.
• Think of expected value as a long
term average.
American Roulette
• At a roulette table in Las Vegas, you will find the following numbers
1 – 36, 0, 00. There are 38 total numbers.
• Let’s say we play our favorite number, 7.
• We place a $1 chip on 7. If the ball lands in the 7 slot we win $35
(net winnings). If the ball lands on any other number we lose our $1
chip.
• What is the expectation of this bet?
• To answer this question we need to know the probability of winning
and losing.
• The probability of winning is 1/38. The probability of losing is 37/38.
• So the expectation is E = $35(1/38) + (-$1)(37/38) =
(35-37)/38 = -2/38 = -$0.053
• What this tells us is that over a long time for every $1 we bet we will
lose $0.053.
• This is an example of a game with a negative expectation. One
should not play games when the expectation is negative.
Example 2
• On the basis of previous experience a librarian knows
that the number of books checked out by a person
visiting the library has the following probabilities:
# of books
Probability
0
1
0.15 0.35
2
3
4
5
0.25
0.15
0.05
0.05
• Find the expected number of books checked out
by a person.
• E = 0(0.15)+1(0.35)+2(0.25)+3(0.15)+4(0.05)+5(0.05)
• E = 0 + 0.35 + 0.50 + 0.45 + 0.20 + 0.25
• E = 1.75
Two dice are rolled
• A player gets $5 if the two dice show the same
number, or if the numbers on the dice are
different then the player pays $1.
• What is the expected value of this game?
• What is the probability of winning $5? ANSWER
6/36 = 1/6.
• What is the probability of paying a $1? ANSWER
5/6.
• Thus E = $5(1/6) + (-$1)(5/6) = 5/6 – 5/6 = 0.
• The Expectation is $0. This would be a fair
game.
Who Wants to be a Millionaire?
• Recall the game show Who Wants to be a
Millionaire? Hosted by Regis Philbin.
• Let’s say you’re at the $125,000 question with
no life-lines. The question you get is the
following
• What Philadelphia Eagles head coach has the
most victories in franchise history?
A. Earle “Greasy” Neale
B. Buddy Ryan
C. Dick Vermeil
D. Andy Reid
More Millionaire
• If you get the question right you will be at
$125,000.
• If you get the question wrong you fall back
to $32,000.
• Your third option is to walk away with
$64,000.
• What to do, what to do?
• Let’s do a mathematical analysis.
Mathematical analysis
• What is the probability of guessing
correctly? ANSWER ¼.
• What is the probability of guessing
incorrectly? ANSWER ¾.
• What is our expectation? ANSWER
E = $125,000(1/4)+$32,000(3/4) =
$55,250. This is less than the $64,000
walk away value.
• Decision: We should walk away.
What if we had a 50/50
• Then 2 of the choices would vanish.
• Now the choices left will be A. Greasy
Neale and D. Andy Reid.
• Now E = (1/2)$125,000 + (1/2)$32,000 =
$78,500 > $64,000.
• We should give it a shot.
• The answer is A. Greasy Neale (for now,
later this season Andy Reid will pass
Neale).