Minds on! Which would you push?
What if you had 2 chances (same button)?
The average case
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There is no right answer.
But…which button will win you more
money in the long run?
On average, what is the payout?
Red: 1M; Green: 50M
An actual contest
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How much would you expect to win?
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What is the average amount won?
What do you need to know?
Expected Value
Section 4.2, 5.1
Learning goal: Calculate the Expected Value of a
probability situation
MSIP / Home Learning: pp. 262 – 265 # 1-3, 5, 7, 9, 18
Calculate Probabilities and Expected Value for your Games
Fair
Expected Value
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The value to which the average of a random
variable’s values tends after many repetitions
Also called the average value
If all outcomes are equally likely, it is the
average of all possible values
Expected Value
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Average of all sides is
3.5
E(X) = 3.5
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Average of all
sections is 2 595
E(X) = 2 595
What if the outcomes are not equally
likely?
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You must factor in the probability as well as
the value
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Each outcome contributes some value to the
overall expectation
For a game with outcomes x1, x2, …, xn, that
have probabilities p1, p2, …, pn
E(X) = p1x1 + p2x2 + … + pnxn
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Multiply the value of every outcome by its
probability and add them up.
Expected/Average Value
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The probability-weighted sum of all possible
values
The long-run average value of an experiment
over many repetitions
Does NOT mean the "most likely value"
May be unlikely or even impossible

The Expected/Average Value of
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One die is 3.5 (impossible)
Two dice is 7 (possible)
Three dice is 10.5 (not possible)
An actual contest
Shoppers Drug Mart - Expected Value
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Example 1 – “Nevada” tickets
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2700 tickets per box - sold for $0.50 each
276 prizes as follows:
 n($100) = 4
 n($50) = 2
 n($25) = 6
 n($5) = 4
 n($1) = 260
What is the probability of buying a winning ticket?
P(win) = 276 / 2 700 = 10.22%
But…would you be equally happy if you won $1 vs. $100?
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How do these factor into the winning scheme?
Example 1 – Probabilities
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First, find the probability of each outcome.
n($100) = 4
P($100) = 4/2700
n($50) = 2
P($50) = 2/2700
n($25) = 6
P(25) = 6/2700
n($5) = 4
P($5) = 4/2700
n($1) = 260
P($1) = 260/2700
Example 1 – Expected Value
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To find the Expected Value, multiply every outcome by
its probability and add them together:
= 0.1481
+ 0.0370 + 0.0556 + 0.0074 + 0.0962
= 0.3443 or $0.34
The Expected Value of a ticket is $0.34. However, the
ticket costs $0.50. So you would expect to lose $0.16 on
every ticket you buy.
Example 2 – Rolling a Die
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Suppose you play a game with a friend where
you roll a fair die.
If the roll is odd, your friend pays you the
amount shown on the die.
If the roll is even, you pay your friend the
amount shown on the die.
How much money would you expect to win or
lose on each roll? Is this a fair game?
Example 2 – Rolling a die (cont’d)
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Odd numbers represent wins and have a positive
value
Even numbers represent losses and have a
negative value
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E(X) = (1)(1/6) + (-2)(1/6) + (3)(1/6) + (-4)(1/6) + (5)(1/6) + (-6)(1/6)
= 1/6 – 2/6 + 3/6 – 4/6 + 5/6 – 6/6
= -3/6
= -1/2 or -$0.50
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So you would expect to lose $0.50 on every play.
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This is not a fair game.
(Not So) Great Expectations!
Which game do you think has the highest return per dollar?
For every dollar spent, you would
expect to LOSE…
Blackjack
Craps
-1.5 cents
-2.8 cents
Crossword -50 cents
Heart and Stroke -65 cents
So looking at Expected Value, the
games look like this:
Blackjack
Craps
E(X) = 0.985
E(X) = 0.972
Crossword E(X) = 0.5
Heart and Stroke E(X) = 0.35