Ralph Vince - Portfolio Mathematics

ch01 JWBK035-Vince February 22, 2007 21 : 43 Char Count= 0

8 THE HANDBOOK OF PORTFOLIO MATHEMATICS

If you bet$1 on one number in roulette (American double-zero), you

would expect to lose, on average, 5.26 cents per roll.If you bet$5, you

would expect to lose, on average, 26.3 cents per roll.Notice howdiffer-

ent amounts bet have different mathematical expectations in terms of

amounts, but the expectation as apercentof the amount bet is always the

same.

The player’s expectation for a series of bets is the total of the ex-

pectations for the individual bets.Soif you play$1 on a numberin

roulette, then$10 on a number, then$5 on a number, your total expectation

is:

ME=(−.0526)∗ 1 +(−.0526)∗ 10 +(−.0526)∗ 5

=−. 0526 −. 526 −. 263

=−. 8416

You would therefore expect to lose on average84.16 cents.

Thisprinciple explains why systems that try to change the size of

their bets relative to how many wins or losses have been seen (assuming

anindependent trials process) are doomed to fail.The sum of negative-

expectation betsis always a negative expectation!

EXACT SEQUENCES, POSSIBLE OUTCOMES,

AND THE NORMAL DISTRIBUTION

We have seen how flippingone coingives us a probability statement with

two possible outcomes—heads or tails.Our mathematical expectation

would be the sum of these possible outcomes.Now let’sflip two coins.

Here the possible outcomes are:

Coin 1 Coin 2 Probability

HH.25

HT .25

TH.25

TT.25

This can also be expressed as there beinga 25% chance ofgettingboth

heads, a 25% chance ofgettingboth tails, and a 50% chance ofgettinga

head and a tail.In tabular format: