ch01 JWBK035-Vince February 22, 2007 21 : 43 Char Count= 0
22 THE HANDBOOK OF PORTFOLIO MATHEMATICS
our example we divide the approximate total pool of$2,000 by 1,000, the
total number of combinations, to obtain an average bet per combination
of$ 2.
Now we figure the total amount bet on the number we want to play.
Here we would needinsideinformation.The purpose hereis not to show
how to win at numbers or any othergamblingsituation, but rather to show
how to think correctlyin approachingagiven risk/reward situation.This
will be made clearer as we continue with theillustration.For now, let’sjust
assume we canget thisinformation.Now,if we know what the average
dollar betis on any number, and we know the total amount bet on the
number we want to play, we simply divide the average bet by the amount
bet on our number.Thisgives us the ratio of what our bet sizeis relative to
the average bet size.
Since the pool can be won by any number, and since the poolis really
the average bet times all possible combinations,it stands to reason that
naturally we want our bet to be relatively small compared to the average
bet.Therefore,ifthis ratiois1.5,it means simply that the average bet on a
numberis1.5times the amount bet on our number.
Now this can be convertedinto an actual mathematical expectation.
We take this ratio and multiplyit by the quantity (1−takeout) where the
takeoutis the pari-mutuel vigorish (also known as the amount that the
house skims off the top, and out of the total pool).In the case of numbers,
where the takeoutis 30%, then 1 minus the takeout equals. 7 .Multiplying
our ratioin our example of 1.5times. 7 gives us 1. 05 .As a final step, sub-
tracting1 from the previous step’s answer willgive us the mathematical
expectation,in percent.Since 1. 05 − 1 is 5%, we can expectin our example
situation to make 5% on our money on averageif we make this play over
and over.
Which bringsustoaninterestingproviso here.In numbers, we have
probabilities of 1/1000 or.001 of winning.So,in our example,if we bet
$1 for each of 1,000 plays, we would expect to be ahead by 5%, or$50,if
thegiven parameters as wejust described were always present.Sinceitis
possible to play the number 1,000 times, the mathematical expectationis
possible, too.
But let’s say you try to do this on a state lottery with over 7 million
possible winningcombinations.Unless you have a pool together or a lot of
money to cover more than one number on each drawing,itis unlikely you
will see over 7 million drawingsin your lifetime.Sinceitwill take (on aver-
age) 7 million drawings until you can mathematically expect your number
to have come up, your positive mathematical expectation as we described
itin the numbers exampleis meaningless.You most likely won’t be around
to collect!