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MAT116 MAT116 Chapter 4: Chapter 4:

Expected ValueExpected Value

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4-1: Summation Notation4-1: Summation Notation

Suppose you want to add up a bunch of Suppose you want to add up a bunch of probabilities for events Eprobabilities for events E11, E, E22, E, E33, … E, … E100100..

One way to write it would be:One way to write it would be:

)()(...)()()( 10099321 EPEPEPEPEP

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Summation NotationSummation Notation

Another, more compact way to write it Another, more compact way to write it would be by using summation notation. would be by using summation notation. This same set of probabilities, added This same set of probabilities, added together, can be expressed as follows:together, can be expressed as follows:

100

1

)(i

iEP

i is called the index

Summation symbol

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Summation NotationSummation Notation

To add the first 25 whole numbers up, To add the first 25 whole numbers up, 1+2+3+…+24+25, we would write this:1+2+3+…+24+25, we would write this:

25

1i

i

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Summation NotationSummation Notation

To add the following sum: 3To add the following sum: 322+4+422+…24+…2422+25+2522

25

3

2

i

iNote that the index starts

at 3 instead of 1 to match the sequence of

numbers you are adding

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ExampleExample

Find the value of the following:Find the value of the following:

6

3

25i

i

77

Optional ExamplesOptional Examples

Find the value of the following:Find the value of the following:

5

1

17

3

6

0

211ikj

j ik

88

Optional ExamplesOptional Examples

Write the following in summation Write the following in summation notation:notation:

– 3(2)3(2)22 + 3(3) + 3(3)22 + 3(4) + 3(4)22 + … + 3(28) + … + 3(28)22 + + 3(29)3(29)22

– FF(0.5)+F(1.5)+F(2.5)+F(3.5)+…+F(10.5)(0.5)+F(1.5)+F(2.5)+F(3.5)+…+F(10.5)– 3(2)3(2)22 - 3(3) - 3(3)22 + 3(4) + 3(4)22 – 3(5) – 3(5)22 … + 3(20) … + 3(20)22 - -

3(21)3(21)22

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4-2: Sums and Probability4-2: Sums and Probability

Summation Notation will come in Summation Notation will come in handy when we want to add the handy when we want to add the probabilities of several events at probabilities of several events at once.once.

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4-3: Excel and Sums4-3: Excel and Sums

Find the value of the following using ExcelFind the value of the following using Excel Note how hard this would be to write our Note how hard this would be to write our

or compute by hand!or compute by hand!

25

7

2 102i

i

1111

4-4: Random Variables4-4: Random Variables A A random variablerandom variable is a variable whose is a variable whose

value can change. In the context of value can change. In the context of probability, it is usually the numerical probability, it is usually the numerical outcome of some random trial or outcome of some random trial or experiment.experiment.

For example, throwing a die has an For example, throwing a die has an associated random variable. Let V be the associated random variable. Let V be the number that comes up on the die. The number that comes up on the die. The outcome, and one of the members of outcome, and one of the members of {1,2,3,4,5,6} is random and so V is a {1,2,3,4,5,6} is random and so V is a random variable.random variable.

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NotationNotation

Suppose V is the random variable Suppose V is the random variable just described for throwing a die. We just described for throwing a die. We will often denote probabilities as will often denote probabilities as follows:follows:P(V=1) = 1/6P(V=1) = 1/6

This is the probability that the die comes

up as a 1

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NotationNotation

P(2 < V P(2 < V ≤ 5) = ???≤ 5) = ??? This is the probability that the This is the probability that the

number that comes up on the die is number that comes up on the die is greater than 2 and less than or equal greater than 2 and less than or equal to 5.to 5.

So, what is So, what is P(2 < V P(2 < V ≤ 5)≤ 5)

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ExampleExample

Let T be the random variable that Let T be the random variable that gives the total of rolling two dice.gives the total of rolling two dice.

What is P(T > 7)?What is P(T > 7)? What is P(4 < T What is P(4 < T ≤ 10)?≤ 10)?

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4-5: Expected Value4-5: Expected Value The Expected Value of a Random Variable The Expected Value of a Random Variable

is the predicted average of all outcomes of is the predicted average of all outcomes of a very large number of trials or random a very large number of trials or random experiments.experiments.

It is the value you expect to get (as an It is the value you expect to get (as an average) and may not actually be equal to average) and may not actually be equal to any of the outcomes that are possible in any of the outcomes that are possible in your experiment. your experiment.

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ExampleExample

if there are 100 slips of paper in a hat (50 if there are 100 slips of paper in a hat (50 with 1 written on them and 50 with 0 with 1 written on them and 50 with 0 written on them), what is the average written on them), what is the average value of a slip you pull out of the hat if value of a slip you pull out of the hat if you pull out “enough” slips of paper?you pull out “enough” slips of paper?

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Expected ValueExpected Value

Suppose you have 60 plastic markers in Suppose you have 60 plastic markers in a box. 20 are marked with as $3, 20 are a box. 20 are marked with as $3, 20 are marked as $4, and 20 are marked as $5.marked as $4, and 20 are marked as $5.

If you randomly choose one of the If you randomly choose one of the markers out of the bag many many markers out of the bag many many times, what is the average (expected times, what is the average (expected value) of such an action? How can you value) of such an action? How can you find the answer without doing any find the answer without doing any computations?computations?

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ExampleExample

Now change the problem so it reads like Now change the problem so it reads like this? Suppose you have 60 plastic this? Suppose you have 60 plastic markers in a box. 20 are marked with markers in a box. 20 are marked with as $3, 10 are marked as $4, and 30 are as $3, 10 are marked as $4, and 30 are marked as $5.marked as $5.

Do you think the expected value will be Do you think the expected value will be the same as before? Smaller? Larger? the same as before? Smaller? Larger? Why?Why?

HOW WOULD YOU FIND SUCH A VALUE?HOW WOULD YOU FIND SUCH A VALUE?

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Definition of Expected ValueDefinition of Expected Value If If XX is a random variable, then is a random variable, then E(X),E(X),

, , and and µµXX can all represent the expected can all represent the expected value of value of XX

If there are If there are nn different numerical different numerical outcomes of a trial, the formula for outcomes of a trial, the formula for Expected Value is:Expected Value is:

where where xx is each possible value of the is each possible value of the random variable, and random variable, and pp is the is the probability of each outcome occurring.probability of each outcome occurring.

nn pxpxpxxpXE ...)( 2211

X

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What does this mean?What does this mean?

Note that each value of the random variable Note that each value of the random variable gets multiplied by its corresponding probability.gets multiplied by its corresponding probability.

So, if a the probability of a particular outcome is So, if a the probability of a particular outcome is large, then it gets multiplied by a larger value. large, then it gets multiplied by a larger value. Hence, it will play a larger role in the final Hence, it will play a larger role in the final expected value result. We say that it is weighted expected value result. We say that it is weighted more heavily.more heavily.

Likewise, an outcome with only a small Likewise, an outcome with only a small probability of happening gets multiplied by a probability of happening gets multiplied by a much smaller value and so it is weighted much much smaller value and so it is weighted much less.less.

nn pxpxpxxpXE ...)( 2211

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Back to our ExampleBack to our Example Now change the problem so it reads like Now change the problem so it reads like

this? Suppose you have 60 plastic this? Suppose you have 60 plastic markers in a box. 20 are marked with markers in a box. 20 are marked with as $3, 10 are marked as $4, and 30 are as $3, 10 are marked as $4, and 30 are marked as $5.marked as $5.

Start by building a probability table that Start by building a probability table that includes columns for the random includes columns for the random variable, its corresponding probability, variable, its corresponding probability, and the product of the two. Each row of and the product of the two. Each row of the table will correspond to a single the table will correspond to a single outcome of the random variable.outcome of the random variable.

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Continuing our ExampleContinuing our Example

xx pp x*px*p

$3$3

$4$4

$5$5

Expected Expected Value=Value=

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ExampleExample

Let Let §§ be the sample space be the sample space represented by all possible outcomes represented by all possible outcomes of tossing three coins on a table.of tossing three coins on a table.

Let X = the number of heads that Let X = the number of heads that occur in a trial (of tossing the three occur in a trial (of tossing the three coins).coins).

What is the expected value of X?What is the expected value of X?

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Group Activity (Time allowing)Group Activity (Time allowing)

Let Let §§ be the sample space be the sample space represented by all possible outcomes represented by all possible outcomes of tossing four coins on a table.of tossing four coins on a table.

Let X = the number of heads that Let X = the number of heads that occur in a trial (of tossing the four occur in a trial (of tossing the four coins).coins).

What is the expected value of X?What is the expected value of X?

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ExampleExample Suppose your local church decides to Suppose your local church decides to

raise money by raffling a microwave raise money by raffling a microwave worth $400. A total of 2000 tickets worth $400. A total of 2000 tickets are sold at $1 each. Find the are sold at $1 each. Find the expected value of winning for a expected value of winning for a person who buys 1 ticket in the person who buys 1 ticket in the raffle. raffle.

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ExampleExample

A 27-year old woman decides to pay A 27-year old woman decides to pay $156 for a one-year life-insurance $156 for a one-year life-insurance policy with coverage of $100,000. policy with coverage of $100,000. The probability of her living through The probability of her living through the year is 0.9995 (based on data the year is 0.9995 (based on data from the US Dept of Health and AFT from the US Dept of Health and AFT Group Life Insurance). What is her Group Life Insurance). What is her expected value for the insurance expected value for the insurance policy. (Ans: -$106)policy. (Ans: -$106)

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ExampleExample

When you give a casino $5 bet on When you give a casino $5 bet on the number 7 in roulette, you have a the number 7 in roulette, you have a 1/38 probability of winning $175 1/38 probability of winning $175 (including your $5 bet) and 37/38 (including your $5 bet) and 37/38 probability of losing $5. What is your probability of losing $5. What is your expected value? In the long run, how expected value? In the long run, how much will you lose for each dollar much will you lose for each dollar bet?bet?

(Ans: E(X) = -$0.26316)(Ans: E(X) = -$0.26316)

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ExampleExample

Suppose you insure a $500 iPod from Suppose you insure a $500 iPod from defects by paying $60 for two years defects by paying $60 for two years of coverage. If the probability of the of coverage. If the probability of the unit becoming defective in that two-unit becoming defective in that two-year period is 0.1, what is the year period is 0.1, what is the expected value of that insurance expected value of that insurance policy?policy?

Ans: -$10Ans: -$10

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Recall my client, John SandersRecall my client, John Sanders

From Chapter 2:From Chapter 2:

Number of successful loans with 7 Number of successful loans with 7 years of experience = 105 (vs 134)years of experience = 105 (vs 134)

Number of successful loans with Number of successful loans with bachelor degrees = 510 (vs 644)bachelor degrees = 510 (vs 644)

Number of successful loans during Number of successful loans during normal times = 807 (vs 740)normal times = 807 (vs 740)

Initial recommendation: forecloseInitial recommendation: foreclose

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From chapter 3:From chapter 3:

P(S) = 46.4% P(S) = 46.4%

P(loan with 7 years of experience will P(loan with 7 years of experience will be successful) =43.9%be successful) =43.9%

P(loan with bachelor’s degree will be P(loan with bachelor’s degree will be successful) = 44.2%successful) = 44.2%

P(loan in normal times will be paid P(loan in normal times will be paid back) = 52.2%back) = 52.2%

Initial recommendation: foreclosureInitial recommendation: foreclosure

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Note though…Note though…

P(a loan with 7 years of experience) = P(a loan with 7 years of experience) = 7.36%7.36%

P(a loan with bachelor’s degree) =53.1%P(a loan with bachelor’s degree) =53.1%P(a loan issued during normal times) P(a loan issued during normal times)

=72.77% =72.77% We have few loans with 7 years of We have few loans with 7 years of

experience….experience….We need to take account the amount of the We need to take account the amount of the

loan, the foreclose value and the default loan, the foreclose value and the default valuevalue

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Focus on the Project: Chapter 4Focus on the Project: Chapter 4

Let Let SS be the event that an attempted be the event that an attempted loan workout is successfulloan workout is successful

Let Let FF be the event that an attempted be the event that an attempted loan workout failsloan workout fails

Let Let ZZ be the random variable that be the random variable that gives the amount of money that gives the amount of money that Acadia Bank receives from a future Acadia Bank receives from a future loan workout.loan workout.

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Question of InterestQuestion of Interest

Expected value of a loan workout : if Expected value of a loan workout : if the loan workout is done many the loan workout is done many times, this is the average value we times, this is the average value we expect to get from such a workout. expect to get from such a workout.

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Focus on the ProjectFocus on the Project We can use the probability of failure We can use the probability of failure

and success to find a preliminary and success to find a preliminary estimate for the expected value of Zestimate for the expected value of Z

Recall that P(S) = 0.464 and P(F) = Recall that P(S) = 0.464 and P(F) = 0.5360.536

E(Z) = f * P(S) + d * P(F)E(Z) = f * P(S) + d * P(F) where f = full amount of loanwhere f = full amount of loan d = default value of loand = default value of loan

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Back to my client, John…Back to my client, John…

Expected value of his 4M loan=Expected value of his 4M loan=

4M * 0.464 + 0.250M*0.536 4M * 0.464 + 0.250M*0.536

=1.99M=1.99M

In the long run, we expect to get In the long run, we expect to get 1.99M from working out the loan. 1.99M from working out the loan.

The expected value of a workout is The expected value of a workout is lower than the foreclose value of lower than the foreclose value of 2.1M2.1M

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Looking at the other expected Looking at the other expected values:values:

Expected value of a workout that matches Expected value of a workout that matches my client’s years of experience = my client’s years of experience = 4*0.44+0.25*0.56=1.9M4*0.44+0.25*0.56=1.9M

Expected value of a workout that matches Expected value of a workout that matches my client’s level of my client’s level of education=4*0.44+0.25*0.56 =1.9Meducation=4*0.44+0.25*0.56 =1.9M

Expected value of a workout that matches Expected value of a workout that matches the economic times = the economic times = 4*0.52+0.25*0.48=2.2M4*0.52+0.25*0.48=2.2M

Two out of three are lower than my Two out of three are lower than my foreclosure value of 2.1Mforeclosure value of 2.1M

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Looking at expected values of each Looking at expected values of each bank:bank:

For BR bank:For BR bank: 4*0.45+0.25*0.55=1.94M4*0.45+0.25*0.55=1.94MFor Cajun bank:For Cajun bank: 4*0.44+0.25*0.56=1.9M4*0.44+0.25*0.56=1.9MFor Dupont bank:For Dupont bank: 4*0.49+0.25*0.51=2.09M4*0.49+0.25*0.51=2.09MAll of these are lower than my foreclose All of these are lower than my foreclose

value of 2.1M.value of 2.1M.Recommendation: forecloseRecommendation: foreclose

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Focus on the ProjectFocus on the Project

Do Parts 2b and 2c of Project 1 Do Parts 2b and 2c of Project 1 Specifics section of the Project 1 Specifics section of the Project 1 materials.materials.

Update your written report to reflect Update your written report to reflect this new information. this new information.