1
Welcome to St. Petersburg!
•
Game setup
We have a fair coin (come up “heads” with p = 0.5)
Let n = number of coin flips before first “tails”
You win $2
n
•
How much would you pay to play?
•
Solution
Let X = your winnings
E[X] =
I’ll let you play for $1 million.
..
but just once!
Takers?
0
1
3
4
2
3
1
2
0
1
2
2
1
...
2
2
1
2
2
1
2
2
1
2
2
1
i
i
i
0
2
1
i
Breaking Vegas
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Consider even money bet (e.g., bet “Red” in roulette)
p = 18/38 you win $Y, otherwise (1 – p) you lose $Y
Consider this algorithm for one series of bets:
1.
Y = $1
2.
Bet Y
3.
If Win, stop
4.
if Loss, Y = 2 * Y, goto 2
Let Z = winnings upon stopping
E[Z]
Expected winnings ≥ 0.
Use algorithm infinitely often!
...
)
1
2
4
(
38
18
38
20
)
1
2
(
38
18
38
20
1
38
18
2
1
38
20
1
1
38
18
38
20
38
18
2
2
38
18
38
20
0
1
1
0
i
i
i
j
j
i
i
i
Vegas Breaks You
•
Why doesn’t everyone do this?
Real games have maximum bet amounts
You have finite money
o
Not be able to keep doubling bet beyond certain point
Casinos can kick you out
•
But, if you had:
No betting limits, and
Infinite money, and
Could play as often as you want.
..
•
Then, go for it!
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 Spring '09
 Variance, Probability theory, Bernoulli random variable, Bernoulli trial, Jacob Bernoulli

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