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Mathematical ExpectationMathematical Expectation

Ernesto Diaz, Mathematics Department

Redwood High School

Mathematical Expectation

What is it?An average.

Example: You buy tires rated for 60,000 miles. This is the expected life of the tires. Of course, your experience may differ depending on driving habits and random chance.

Mathematical Expectation

What is it used for?Typically, as a tool for decision making.

Definition: E = a1p1 + a2p2 + … + anpn

The ai’s are values of individual outcomes.The pi’s are probabilities of those outcomes.

An (contrived) Example

Given: Bet on a 4-horse race, with pay-off $100 if Horse 1 wins and $50 if horse 2 wins. Nothing if horses 3 or 4 win. What is the expected value of your bet?

Note: a1 = $100, a2 = $50, a3=a4=$0

p1 = p2 = p3 = p4 = 1/4 (assumed)

Answer:

E = $100(1/4) + $50(1/4) + $0(1/4) + $0(1/4)

= $25 + $12.50

= $37.50 an unusually good bet!

Expected Value

The symbol P1 represents the probability that the first event will occur, and A1 represents the net amount won or lost if the first event occurs.

1 1 2 2 3 3 ... n nE P A P A P A P A

Example Teresa is taking a multiple-choice test in which

there are four possible answers for each question. The instructor indicated that she will be awarded 3 points for each correct answer and she will lose 1 point for each incorrect answer and no points will be awarded or subtracted for answers left blank. If Teresa does not know the correct answer

to a question, is it to her advantage or disadvantage to guess?

If she can eliminate one of the possible choices, is it to her advantage or disadvantage to guess at the answer?

Solution Expected value if Teresa guesses.

1(guesses correctly)

4P

3(guesses incorrectly)

4P

1 3Teresa's expectation = (3) ( 1)

4 43 3

04 4

Solution continued—eliminate a choice

1

(guesses correctly)3

P

2(guesses incorrectly)

3P

1 2Teresa's expectation = (3) ( 1)

3 32 1

13 3

Example: Winning a Prize When Calvin Winters attends a

tree farm event, he is given a free ticket for the $75 door prize. A total of 150 tickets will be given out. Determine his expectation of winning the door prize.

1 149Expectation = (75) (0)

150 1501

2

Example When Calvin Winters attends a

tree farm event, he is given the opportunity to purchase a ticket for the $75 door prize. The cost of the ticket is $3, and 150 tickets will be sold. Determine Calvin’s expectation if he purchases one ticket.

Solution

Calvin’s expectation is $2.49 when he purchases one ticket.

1 149Expectation = (73) ( 3)

150 15073 447

150 150374

1502.49

Fair Price

Fair price = expected value + cost to play

Example Suppose you are playing a game in which you

spin the pointer shown in the figure, and you are awarded the amount shown under the pointer. If is costs $10 to play the game, determine

a) the expectation of the person who plays the game.

b) the fair price to play the game.

$10

$10

$2

$2

$20

$15

Solution

$0

3/8

$10

$10$5$8Amt. Won/Lost

1/81/83/8Probability

$20$15$2Amt. Shown on Wheel

3 3 1 1Expectation = ( $8) ($0) ($5) ($10)

8 8 8 824 5 10

08 8 89

1.125 1.138

Solution Fair price = expectation + cost to

play = $1.13 + $10

= $8.87

Thus, the fair price is about $8.87.

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Why the House WinsGame Expected value

for $1 bet•Baccarat

•Blackjack

•Craps

•Slot machines

•Keno (eight-spot ticket)

•Average state lottery

-$0.014

-0.06 to + $0.10 (varies with strategies)

-$0.014 for pass, don’t pass, come, don’t come bets only

-$0.13 to ? (varies)

-$0.29

-$0.48

There isn’t a single bet in any game of chance with which you can expectto break even, let alone make a profit