18.8. Really Great Expectations 645
you’ve won, and I’ll pay you generously —how does 25 cents sound? Maybe you’d
rather have $1? How about $10?
Of course there’s a trap here. Let’s calculate your expected winnings.
Suppose your initial run of tails had lengthk. After that, each time a head comes
up, you have to start over and try to getkC 1 tails in a row. If we regard your getting
kC 1 tails in a row as a “failed” try, and regard your having to start over because a
head came up too soon as a “successful” try, then the number of times you have to
start over is the number of tries till the first failure. So the expected number of tries
will be the mean time to failure, which is 2 kC^1 , because the probability of tossing
kC 1 tails in a row is 2 .kC1/.
LetTbe the length of your initial run of tails. SoT Dkmeans that your initial
tosses wereTkH. LetRbe the number of times you repeat trying to beat your
original run of tails. The number of cents you expect to finish with is the number
of cents in my generous payment minus ExŒRç. It’s now easy to calculate ExŒRçby
conditioning on the value ofT:
ExŒRçDX
k 2 NExŒRjT DkçPrŒT DkçDX
k 2 N2 kC^1 2 .kC1/DX
k 2 N1 D1:
So you can expect to pay me an infinite number of cents before winning my
“generous” payment. No amount of generosity can make this bet fair!
We haven’t faced infinite expectations until now, but they just popped up in a
very simple way. In fact this particular example is a special case of an astonish-
ingly general one worked out in Problem 18.23: the expected waiting time forany
random variable to achieve a larger value is infinite.
18.8.2 The St. Petersburg Paradox
One of the simplest casino bets is on “red” or “black” at the roulette table. In each
play at roulette, a small ball is set spinning around a roulette wheel until it lands in
a red, black, or green colored slot. The payoff for a bet on red or black matches the
bet; for example, if you bet $ 10 on red and the ball lands in a red slot, you get back
your original $ 10 bet plus another matching $ 10.
In the US, a roulette wheel has two green slots among 18 black and 18 red slots,
so the probability of red is18=380:473. In Europe, where roulette wheels have
only one green slot, the odds for red are a little better —that is,18=370:486
—but still less than even.
There is a notorious gambling strategy allegedly used against the casino in St.
Peterburg way back in czarist days: bet $ 10 on red, and keep doubling the bet until
a red comes up. This strategy implies that a player will leave the game as a net
winner of $ 10 as soon as the red first appears.