Ralph Vince - Portfolio Mathematics

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

The Random Process and Gambling Theory 13

common misconception that can lead to a great deal of trouble. It is the

end results (not the exact sequences) that conform to the bell curve—the

Normal Distribution,whichis a particular type of probability distribution.

Aninterestingcharacteristic of all probability distributionsis a statistic

known as thestandard deviation.

For the Normal Probability Distributiononasimple binomialgame,

such as the one beingused here for the end results of coinflips, the stan-

dard deviation (SD)is:

D=N∗

√

P∗(1−P)

N

(1.04)

where: P=Probability of the event (e.g., result of heads).

N=Number of trials.

For 10 coin tosses (i.e.,N=10):

SD= 10 ∗

√

. 5 ∗(1−.5)/ 10

= 10 ∗

√

. 5 ∗. 5 / 10

= 10 ∗

√

. 25 / 10

= 10 ∗. 158113883

= 1. 58113883

The center line of a distributionis the peak of the distribution.In the

case of the coin toss the peakis at an even number of heads and tails.So for

a 10-toss sequence, the center line would be at 5 heads and 5 tails.For the

Normal Probability Distribution, approximately 68.26% of the events will

be+or−1 standard deviation from the center line, 95.45% between+and

−2 standard deviations from the center line, and 99.73% between+and

−3 standard deviations from the center line (see Figure 1.2).Continuing

with our 10-flipcoin toss, 1 standard deviation equals approximately 1. 58.

We can therefore say of our 10-coinflip that 68% of the time we can expect

to have our end result be composed of 3.42 (5− 1 .58) to 6.58 (5+ 1 .58)

beingheads (or tails).Soif we have 7 heads (or tails), we would be beyond

1 standard deviation of the expected outcome (the expected outcome be-

ing5 heads and 5 tails).

Hereis anotherinterestingphenomenon.Noticein our coin-toss exam-

ples that as the number of coins tossedincreases, the probability ofgetting

an even number of heads and tails decreases.With two coins the probabil-

ity ofgettingH1T1 was. 5 .At four coins the probability ofgetting50% heads

and 50% tails dropped to. 375 .At sixcoinsit was.3125, and at 10 coins. 246.

Therefore, we can state thatas the number of events increases, the prob-

ability of the end result exactly equaling the expected value decreases.