Ralph Vince - Portfolio Mathematics

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

The Random Process and Gambling Theory 15

FIGURE 1.3 The random process: Results of 60 coin tosses, with 1 and 2 stan-

dard deviations in either direction

further you will be from your expected value (in terms of units won or

lost).However, as Nincreases, the standard deviation as a percent of N de-

creases.This means thatthe longer you play, the closer to your expected

value you will be as a percent of the total action (N).Thisisthe“Law of

Averages”presentedinits mathematically correct form.In other words,if

you make a longseries of bets, N, where T equals your total profit or loss

and E equals your expected profit or loss, then T/N tends towards E/N as N

increases.Also, the difference between E and Tincreases as Nincreases.

In Figure 1.3 we observe the random processin action with a 60-coin-

tossgame.Also on this chart you will see the lines for+and−1 and 2

standard deviations.Notice how they bendin, yet continue outward for-

ever.This conforms with what wasjust said about the Law of Averages.

The House Advantage

Now let us examine what happens when thereis a house advantagein-

volved.Again, refer to our coin-toss example.We last saw 60 trials at an

even or“fair”game.Let’s now see what happensif the house has a 5%

advantage.An example of such agame would be a coin toss whereifwe

win, we win$1, butif we lose, we lose$ 1. 10.

Figure 1.4 shows the same 60-coin-tossgame as we previously saw,

only thistime thereis the 5% house advantageinvolved.Notice how,in