Transcript ppt

Chapter 20
The House Edge: expected Values
Chapter 15
1
Choose which one to play ?
Multistate Lotteries
Roulette
Have enormous jackpots
but very small
probabilities of winning.
Have a much more larger
probabilities of winning
but with small jackpots.
Expected values provide one way to
compare these two games.
Chapter 15
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The Tri-State Daily Numbers
You pay $1 and choose a three-digit number.
The state choose a three-digit winning number
at random and pays you $500 if your number is
chosen.
Outcome :
$0
$ 500
Probability
0.999
0.001
What are your average winnings?
Chapter 15
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The Tri-State Daily Numbers
The ordinary average of the two possible
outcomes $0 and $500 is $250.
It makes no sense as the average winning
because $500 is much less likely than $0.
500*0.001 + 0*0.999 = 0.5
Chapter 15
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Expected Values
The expected value of a random phenomenon
that has numerical outcomes is found by
multiplying each outcome by its probability and
then adding all the products.
EV  a1 p1  a 2 p 2    a k p k
ai represents a possible amount,
pi represents the probability of getting this amount
Chapter 15
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Buy a raffle
A raffle has a first prize of $1000, a second prize
of $500 and ten third prizes of $50 each. There
are 500 tickets in all.
Outcome
Probability
Amount
A*P
First Prize
1/500 = 0.002
1000
1000*0.002=2
Second Prize
1/500 = 0.002
500
500*0.002=1
Third Prize
10/500 = 0.02
50
50*0.02 = 1
Total = 4
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The EV is average amount that you
will win per play
if you play the game over and over many times.
Suppose that you buy all the tickets. Then you
are guaranteed to win all the prizes. You win a
total of $2000.
That averages out to $2000/500 = $4 per ticket.
Chapter 15
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The EV is the fair price that you
should pay to play the game
If you pay $4 per ticket then you will exactly
break even.
If the organization that is putting on the lottery
charges $5 per ticket, then they will make a
profit (at the customers’ expense) of $1 per
ticket, or $500 in all.
Chapter 15
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Deal or No Deal?
(1) You choose one of four sealed cases; one contains
$1,000, and the others are empty. If you open your
case, you have a 25% chance to win $1,000 and a
75% chance of getting nothing (winning $0).
(2) Or, you can sell your unopened case for $240,
giving you a 100% chance of winning $240.
• First option (open your case):
EV=($1000)(0.25) + ($0)(0.75) = $250
• Second option (sell your case):
EV=$240, no variation.
• Make a Decision: Will you open or sell your case?
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Deal or No Deal?
(1) A 25% chance to win $1,000 and a 75%
chance of getting nothing. EV=$250
(2) A gift of $240, guaranteed. EV=$240
• If choosing for ONE trial:
– option (1) will maximize potential gain ($1000)
– option (1) will also minimize potential gain ($0)
– option (2) guarantees a gain
• If choosing for MANY (500?) trials:
– option (1) will maximize expected gain
(will make more money in the long run)
Chapter 20
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The Law of Large Numbers
If a random phenomenon with numerical
outcomes is repeated many times
independently, the mean of the actually
observed outcomes approaches the Expected
Value.
Chapter 15
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The Law of Large Numbers
Gambling
• The “house” in a gambling operation is not
gambling at all.
– the games are defined so that the gambler has a
negative expected gain per play
– each play is independent of previous plays, so the law of
large numbers guarantees that the average winnings of
a large number of customers will be close the (negative)
expected value.
No matter how you load the dice or stack the cards,
As long as enough bets are placed, the Law of Large
Numbers guarantees the profit of casino.
Chapter 20
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