Expected Value

One of the reasons for messing with statistics is to help make decisions.  Expected value is a useful concept, it is the average outcome of a decision (not the outcome of an individual decision).

 An office lottery is a good example.  Each week 10 people buy a $1 ticket which provides a $10 prize fund.  On Friday, one ticket is picked at random and the holder of that ticket gets a prize of $10.  Lets take the fun out of it and add some maths.  Buying a lottery ticket is a decision with two outcomes.  The probability of winning is 0.1 and the probability of losing is 0.9.  This can be expressed in a table which in turn can be used to calculate the average outcome (a process similar to that used to calculate an average from a frequency table).

Outcome	Probability	Result	Average
Win	0.1	+9.00	+9.00
Loose	0.9	-1.00	-9.00
Total	1.0		0.00

Whilst the winner gets $10, they have paid $1 for the ticket, hence their net gain is $9.  Over time, the players are simply passing $10 around the office, the expected value of the ticket is zero.

At some point, someone decides the office lottery should do something useful and fund the purchase of a bunch of flowers costing $4, thus reducing the prize fund to $6.  The table below shows the new outcomes:

Outcome	Probability	Result	Average
Win	0.1	+5.00	+5.00
Loose	0.9	-1.00	-9.00
Total	1.0		-4.00

Formerly the expected value is average of the probability  distribution, which can be expressed as:

e = ∑ x p

or

e = ∑ x f/n (where f is the frequency and n population size)

This can illustrated with a binomial distribution where with N=5 and P=0.3.

Outcome	Probability	Average
0	0.168	0.000
1	0.360	0.360
2	0.309	0.617
3	0.132	0.397
4	0.028	0.113
5	0.002	0.012
Total	1.000	1.500

It can be seen that the sum of the probability of the outcomes is 1.00 (i.e. we've got data for all options) and that the average is 1.5, this compares to the theoretical average fro the binomial distribution which is:

e = mean = np

Whilst the concept of expected value derives from a probability distribution, it is often clearer to view the data in the form of a tree structure which makes it possible to display a wider range and complexity of outcomes.  The example below describes the experience of a hypothetical publishing business.

The business advances its authors $50k, 40% of them fail to deliver a manuscript.  Of the 60% who do produce some thing, In 30% of cases, the profits are poor, and 20% are good, typically most produce a profit of $40k.

Page updated: 25-Sep-2007

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