Mathematics for Computer Science

18.8. Really Great Expectations 647

So sure enough,

ExŒCçWWDExŒ

X

i 2 ZC

CiçD10: (18.25)

But since ExŒCiçD 0 ,

X

i 2 ZC

ExŒCiçD

X

i 2 ZC

0 D0: (18.26)

It seems that (18.26) and (18.25) contradict each other, but they don’t. The apparent
contradiction comes from applying infinite linearity to conclude

False Claim.

Ex

2

4

X

i 2 ZC

Ci

3

(^5) D

X

i 2 ZC

ExŒCiç:

But this is a case where the convergence conditions required for infinite linearity
don’t hold. Even though the left hand sum converges (to 10) and the right hand sum
converges (to 0), the infinite linearity Theorem (17.5.5) requires that the sum of
expectations ofabsolute valuesconverges. That is, infinite linearity would follow
if the sum X

i 2 ZC

ExŒjCijç (18.27)

converged. But

ExŒjCijçD.j 10 
 2 i^1 j/
PrŒ1st red inith spinç

C.j 10 
 2 i^1 j/
PrŒ1st red afterith spinç

C 0 
PrŒ1st red before theith spinç

D.10
 2 i^1 /
 2 .i/C.10
 2 i^1 /
 2 .i/C 0 D10;

so the sum (18.27) diverges —rapidly.
Probability theory truly leads to this absurd conclusion: a game entailing an
unbounded number of fair bets may not be fair in the end. In fact, even against an
unfairwheel, as long as there is some fixed positive probability of red on each spin,
you are certain to win $ 10 playing the St. Petersburg strategy!
This brings us to the second thing that’s wrong here: you may wind up losing a
lot of money before you catch up with your net win of $10. LetLbe the number of
dollars you need to have in order to keep betting until the wheel finally spins red.
If red first comes up on theith spin, thenLwould equal

10.1C 2 C 4 C


C 2 i/D10.2iC^1 1/