31 Mar 2012

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Chapter 20

The House Edge: Expected Values

Expected Values

The expected value of a random phenomenon that has numerical outcomes is found by

multiplying each outcome by its probability and then adding all the products

Expected value = a1p1 + a2p2+ …+akpk

An expected value is an average of the possible outcomes, but it is not an ordinary average in

which all outcomes get the same weight

The idea of expected value as an average applies to random outcomes other than games of chance

It is used to describe the uncertain return from buying stocks or building a new factory

The Law of Large numbers

According to the law of large numbers, if a random phenomenon with numerical outcomes is

repeated many times independently, the mean of the actually observed outcomes approaches the

expected value

In many independent repetitions, the proportion of each possible outcome will be close to its

probability, and the average outcome obtained will be closed to the expected value (express the

long-run regularity of chance events)

How large is a large number\/

The law of large numbers says that the actual average outcome of many trials gets closer to the

expected value as more trials are made

Variability of the random outcomes – how many trails are needed to guarantee an average

outcome close to the expected value