QMB s13 PP Sup 1 Expected Value

7/27/2019 QMB s13 PP Sup 1 Expected Value

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Expected Value is what should happen, onaverage, per trial if there are many trials.

This is related to the Law of Large Numbers.


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Expected Value = EVIf xs represent all outcomes and p(x) represents

the probability of x happening:

EV = [xp(x)]

This formula may look confusing, but it isactually really easy to use.


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EV = [xp(x)]

is the Greek capital letter sigma.It means take the sum.All you have to do is multiply every x times its

p(x) and then add up all those products.That is how you find expected value EV!


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Lets say I am going to play a game where I spina wheel with five equal parts.

On three spots I lose $5, one spot I gain $10,and one spot I gain $1.

What is the expected value of my bet?


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Each part of the wheel has a 1/5 = 0.2probability of landing.

Since the lose $5 takes up three partsP( $5) = 3/5 = 0.6Gain $10 takes up one part.P(+$10) = 1/5 = 0.2Same for the gain $1P(+$1) = 1/5 = 0.2


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Lets make a table:

x p(x) x p(x)

Lose $5 5 0.6 5 0.6 = 3.0Gain $10 10 0.2 10 0.2 = 2.0Gain $ 1 1 0.2 1 0.2 = 0.2

0.8EV = $ 0.80


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What does EV = $ 0.80 mean?

Does it mean that every time you play youwill lose 80 cents?

That cant be it because losing 80 cents is

not an outcome.


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Does it mean you will lose every time you play?

It cant mean that because you can very easilywin this game.


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EV = $ 0.80

It means that if you play many times, onaverage you will lose approximately 80cents for every time you played.


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If you play 100 times, you can expect tolose

100($0.80) = $80

Is this guaranteed?Anything could happen. But, like the Law of

Large numbers, the more trials, the moreaccurate the EV.


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Although gambling is the obvious example,expected value is very useful and can be used

in many important ways.

Lets say you are a banker that is completing aninvestment.


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Your consultants tell you that with thisinvestment there is a 70% chance of

profiting $10,000, a 20% of losing$30,000, and a 10% chance of breakingeven.

What is the expected value of the investment?Should the bank do it?


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Lets make a tablex p(x) x p(x)

10,000 0.7 10,000 0.7 = 7,00030,000 0.2 30,000 0.2 = 6,0000 0.1 0 0.1 = 0

1,000EV = $1,000


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Be careful what this means. It doesnt meanthat the bank will profit $1,000 dollars on this

investment.

Every investment is different, so the bank canonly make this investment once. It cant havemanytrials with this investment.


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What investors can do is find the expectedvalue of all their investments and only make

investments that have positive expectedvalues.

They will profit from some and lose some, but,in the end, if all EVs are positive, they shouldmake an overall profit.


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Maybe some of the bankers should have paidattention when they took QMB 3600!

If they only made investments that had apositive EV, then we would not be in thisfinancial crisis!


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Expected value does not always have to be aboutmoney.

Lets say a sports apparel store sells Rays baseball

caps.The manager estimates that if the Rays win the

World Series, the store will sell 500 caps. If theRays do not win the world series, the store will

sell only 100 caps.The paper gives the Rays a 20% chance of winning

the World Series. Try to find the EV!


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Lets make a chart.x p(x) x p(x)

win WS 500 0.2 500 0.2 = 100not win WS 100 0.8 100 0.8 = 80 180

The EV = 180 caps


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A principle of a high school has a concern. TheRays are in the Word Series and it is tied in

the 9th

inning. He knows that many of hisstudents are watching the game an it isalready midnight (since the games start sodarn late).

He feels that the later the game goes, the lessstudents will make 1st period tomorrowmorning.


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x p(x) x p(x)9th 600 0.3 600 0.3 = 180

10th

500 0.4 500 0.4 = 20011th 300 0.2 300 0.2 = 6012th 200 0.1 200 0.1 = _20

460EV = 460 students


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QMB s13 PP Sup 1 Expected Value