Ralph Vince - Portfolio Mathematics

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

12 THE HANDBOOK OF PORTFOLIO MATHEMATICS

The term“exact sequence”here means the exact outcome of a random

process.The set of all possible exact sequences for agiven situationis

called thesample space.Note that the four-coinflipjust depicted can be

four coins all flipped at once, orit can be one coinflipped four times (i.e.,

it can be a chronological sequence).

If we examine the exact sequence T H H T and the sequence H H T T,

the outcome would be the same for a person flat-betting(i.e., betting1unit

on eachinstance).However, to a person not flat-betting, the end result of

these two exact sequences can be far different.To a flat bettor there are

only five possible outcomes to a four-flip sequence:

4 Heads

3 Heads and 1 Tail

2 Heads and 2 Tails

1 Head and 3 Tails

4Tails

As we have seen, there are 16 possible exact sequences for a four-

coinflip.This fact would concern a person whois not flat-betting.We will

refer to people who are not flat-bettingas“system”players, since thatis

most likely what they are doing—bettingvariable amounts based on some

scheme they think they have worked out.

If you flipacoin four times, you will of course see only one of the

16 possible exact sequences.If you flip the coin another four times, you

will see another exact sequence (although you could, with a probability

of 1/16=.0625, see the exact same sequence).If yougouptoagaming

table and watch a series of four plays, you will see only one of the 16 exact

sequences.You will also see one of the five possible end results.Each exact

sequence (permutation) has the same probability of occurring, that being

. 0625 .But each end result (combination) does not have equal probability
of occurring:

End Result Probability

4 Heads .0625

3 Heads and 1 Tail .25

2 Heads and 2 Tails .375

1 Head and 3 Tails .25

4 Tails .0625

Most people do not understand the difference between exact sequences

(permutation) and end results (combination) and as a result falsely con-

clude that exact sequences and end results are the same thing. This is a