Mathematics for Computer Science

Chapter 18 Deviation from the Mean646

Suppose you had the good fortune to gamble against a fair roulette wheel. Then
whatever your bet on a spin of the wheel, you are equally likely to win or lose,
and your expected win is 0. This also means that the expected win after any given
number of spins remains zero, so even playing the St. Peterburg strategy it seems
your expected win would be 0.
But wait a minute. As long as there is a fixed, positive probability of red appear-
ing on each spin of the wheel, it’scertainthat red will eventually come up. That
is, you can be certain of leaving the casino having won $ 10. This implies that even
against anunfairroulette wheel, your expected win is $ 10 , contradicting the idea
that you can’t expect to win in a game that’s biased against you.
This is paradoxical and something’s obviously wrong here. In fact, there are two
things wrong.
The first thing that’s wrong is the argument claiming that the expectation is 0. It
would be 0 if the number of bets had a fixed bound. If you could only makenbets,
then your expectation in the fair game would be the sum of your expected wins on
each of the bets, namely,n
0 D 0. But there is no such fixed bound, and that
changes things.
To explain this carefully, letCibe the number of dollars won on theith spin. So
CiD 2 i^1 when red comes up for the first time on theith spin, andCiD 2 i^1 ,
when the first red spin comes up after theith spin. We can defineCito be 0 if the
first red comes up before theith spin. This means

ExŒCiçD0:

Also, the total of your winnings is

CWWD

X

i 2 ZC

Ci:

The conclusion that ExŒCçD 10 follows from Total Expectation, conditioning on
the number of spins till a red first occurs. Namely, if the first red occurs on theith
spin, the amount won is

 10 
.1C 2 C 22 C


C 2 i^2 /C 10 
 2 i^1 D10:

Then by Total Expectation,

ExŒCçD

X

i 2 ZC

ExŒCjfirst red onith spinç
PrŒfirst red onith spinç

D

X

i 2 ZC

10 
 2 i

D 10

X

i 2 ZC

2 iD 10 
 1 D10: