Ralph Vince - Portfolio Mathematics

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

The Random Process and Gambling Theory 7

This formulajust described willgive us the mathematical expecta-

tion for an event that can have two possible outcomes.What about situa-

tions where there are more than two possible outcomes?The next formula

willgive us the mathematical expectation for an unlimited number of out-

comes.It will alsogive us the mathematical expectation for an event with

only two possible outcomes such as the 2 for 1 coin tossjust described.

Hence,itis the preferred formula.

Mathematical Expectation=

∑N

i= 1

(Pi∗Ai)(1.03a)

where: P=Probability of winningor losing.

A=Amount won or lost.

N=Number of possible outcomes.

The mathematical expectationis computed by multiplyingeach possible

gain or loss by the probability of thatgain or loss, and then summingthose

products together.

Now look at the mathematical expectation for our 2 for 1 coin toss

under the newer, more complete formula:

Mathematical Expectation=. 5 ∗ 2 +. 5 ∗(−1)

= 1 +(−.5)

=. 5

In such an instance, of course, your mathematical expectationisto

win 50 cents per toss on average.

Suppose you are playingagameinwhich you mustguess one of three

different numbers.Each number has the same probability of occurring

(.33), butif youguess one of the numbers you will lose$1,if youguess

another number you will lose$2, andif youguess the right number you

will win$ 3 .Given such a case, the mathematical expectation (ME)is:

ME=. 33 ∗(−1)+. 33 ∗(−2)+. 33 ∗ 3

=−. 33 −. 66 +. 99

= 0

Consider bettingon one numberin roulette, where your mathematical

expectationis:

ME= 1 / 38 ∗ 35 + 37 / 38 ∗(−1)

=. 02631578947 ∗ 35 +. 9736842105 ∗(−1)

=. 9210526315 +(−.9736842105)

=−. 05263157903