Probability and Statistics
Copyright © Cengage Learning. All rights reserved.
14.4
Expected Value
Copyright © Cengage Learning. All rights reserved.
Objectives
►  Expected Value
►  What Is a Fair Game?
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Expected Value
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Expected Value
Suppose that a coin has probability 0.8 of showing heads. If
the coin is tossed many times, we would expect to get
heads about 80% of the time. Now, suppose that you get a
payout of one dollar for each head. If you play this game
many times, you would expect on average to gain $0.80
per game:
= $1.00 × 0.80 = $0.80
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Expected Value
The reasoning in the example discussed motivates the
following definition.
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Example 2 – Finding Expected Value
In Monte Carlo the game of roulette is played on a wheel
with slots numbered 0, 1, 2, . . . , 36.
The wheel is spun, and a ball dropped in the wheel is
equally likely to end up in any one of the slots. To play the
game, you bet $1 on any number. (For example, you may
bet $1 on number 23.)
If the ball stops in your slot, you get $36 (the $1 you bet
plus $35). Find the expected value of this game.
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Example 2 – Solution
The gambler can gain $35 with probability
$1 with probability .
and can lose
So the gambler’s expected value is
In other words, if you play this game many times, you
would expect to lose 2.7 cents on every dollar you bet (on
average).
Consequently, the house expects to gain 2.7 cents on
every dollar that is bet.
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What Is a Fair Game?
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What Is a Fair Game?
A fair game is game with expected value zero. So if you
play a fair game many times, you would expect, on
average, to break even.
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Example 4 – A Fair Game?
Suppose that you play the following game. A card is drawn
from a deck. If the card is an ace, you get a payout of $10.
If the card is not an ace, you have to pay $1.
(a) Is this a fair game?
(b) If the game is not fair, find the payout amount that
would make this game a fair game.
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Example 4(a) – Solution
In this game you get a payout of $10 if an ace is drawn
(probability ), and you lose $1 if any other card is
drawn (probability ).
So the expected value is
Since the expected value is not zero, the game is not
fair. If you play this game many times, you would expect
to lose, on average, ≈ $0.15, per game.
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Example 4(b) – Solution
cont’d
We want to find the payout x that makes the expected
value 0.
=0
Solving this equation we get x = 12. So a payout of $12 for
an ace would make this a fair game.
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What Is a Fair Game?
Games of chance in casinos are never fair; the gambler
always has a negative expected value (as in Examples 2
and 4(a)).
This makes gambling profitable for the casino and
unprofitable for the gambler.
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