Mathematics for Computer Science
MIT 6.042J/18.062J
Infinite
Expectation
Albert R Meyer,
May 14, 2012
infexp.1
Bet-Doubling Paradox
How to beat an unfair
roulette game:
Albert R Meyer,
May 14, 2012
infexp.2
Bet-Doubling Paradox
How to beat an unfair
roulette game: bet $10 on
Black. If you win, go home
with the $10.
Albert R Meyer,
May 14, 2012
infexp.3
Bet-Doubling Paradox
How to beat an unfair
roulette game: bet $10 on
Black. If you win, go home
with the $10. If you lose,
bet $20.
Albert R Meyer,
May 14, 2012
infexp.4
Bet-Doubling Paradox
How to beat an unfair
roulette game: bet $10 on
Black. If you win, go home
with the $10. If you lose,
bet $20. If you win this
time, go home with
$(20-10) = $10.
Albert R Meyer,
May 14, 2012
infexp.5
Bet-Doubling Paradox
How to beat an unfair
roulette game: bet $10 on
Black. If you lose the $20
bet, then bet $40.
Albert R Meyer,
May 14, 2012
infexp.6
Bet-Doubling Paradox
How to beat an unfair
roulette game: bet $10 on
Black. If you lose the $20
bet, then bet $40. If you
win, go home with
$(40 -20 -10) = $10.
Albert R Meyer,
May 14, 2012
infexp.7
Bet-Doubling Paradox
By continuing to double your
bet, your net win will be $10
when you finally win a bet…
and you are certain to
eventually win, since
Pr[bet > k times] 
(10/19)k
Albert R Meyer,
May 14, 2012
infexp.8
Bet-Doubling Paradox
So paradoxically, you are
certain to win an unfair game.
What’s the hitch: what
bankroll will let you play?
E[$ in last bet] =
Albert R Meyer,
May 14, 2012
infexp.9
Bet-Doubling Paradox
So paradoxically, you are
certain to win an unfair game.
What’s the hitch: what
bankroll will let you play?
E[$ in last bet] =
Albert R Meyer,
May 14, 2012
infexp.10
Bet-Doubling Paradox
So paradoxically, you are
certain to win an unfair game.
What’s the hitch? What
initial stake will let you play?
E[$ in last bet] =
Albert R Meyer,
May 14, 2012
infexp.11
Bet-Doubling Paradox
To be certain to win an
unfair game, you need an
infinite bankroll.
Albert R Meyer,
May 14, 2012
infexp.12
Bet-Doubling Paradox
With a finite bankroll, your
expectation goes back to
being negative—as it should
in an unfair game.
Albert R Meyer,
May 14, 2012
infexp.13
Infinite Expectations
• Bet doubling: win is certain, but
expected last bet is infinite.
• Fair gambler’s ruin: ruin is certain,
but expected to take forever.
• E[# tries to beat 1st try] =
• E[R] <
, but maybe Var[R] =
Albert R Meyer,
May 14, 2012
Lec 14M.14
Coping with
Expectation
Average of repeated trials
does not converge. (No Weak
Law.)
But even if E[R] = , maybe
Albert R Meyer,
May 14, 2012
infexp.15
Coping with
Expectation
Average of repeated trials
does not converge. (No Weak
Law.)
But even if E[R] = , maybe
Albert R Meyer,
May 14, 2012
infexp.16
Coping with
Expectation
Average of repeated trials
does not converge. (No Weak
Law.)
But even if E[R] = ,
analyze rate of divergence.
Albert R Meyer,
May 14, 2012
infexp.17
Moral of the Story
Infinite expectation
comes up regularly in
extreme situations,
but can be coped with
Albert R Meyer,
May 14, 2012
infexp.18