Expected Value

Expected Value

When faced with uncertainties, decisions are usually not based solely on probabilities

A building contractor has to decide whether to bid on a construction job: 20% chance of a $40,000 profit 80% chance of a $9,000 loss

Do we bid on the contract?

Expected Value

The chances for a profit are not very high but we stand to gain more than we stand to lose

How do we combine probabilities and consequences?

Expected Value

Consider the following: A person aged 22 can expect to live 51

more years A married woman can expect to have 2.4

children A person can expect to eat 10.4 pounds

of cheese and 324 eggs in a year

What do we mean we say “expect”?

Expected Value

Mathematical expectation can be interpreted as an average A person aged 22 can expect to live an

average of 51 more years A married woman can expect to have an

average of 2.4 children A person can expect to eat an average

of 10.4 pounds of cheese and 324 eggs in a year

Expected Value

Ex: Suppose there are 1000 raffle tickets. There is a $500 prize for the winning ticket and a consolation prize of $1.00 for all other tickets. How much can a person expect to win playing the raffle?

Expected Value

Sol: Suppose all 1000 tickets are drawn and each person’s winnings was recorded. What would a person’s average winnings be?

499.1

1000

9991

1000

1500

1000

1111500Winnings Average

Notice this is the probability of getting the winning ticket

Notice this is the probability of not getting the winning ticket

Expected Value The last slide tells us several things:

Each amount won has a probability associated with it

The amount won is multiplied by its respective probability

The sum of the products is the expected value

Expected value is a weighted average (if we run the experiment many times, what is the average)

Expected Value

What is a weighted average?

Ex: A student computes his average grade in a course in which he took six exams: 75, 90, 75, 87, 75, and 90. He computes his average score as follows:

826

492

6

907587759075

Expected Value Notice he can also write the same average

as:

The average is the weighted average of the student’s grade, each grade being weighted by the probability the grade occurs

826

290

6

187

6

375

6

290187375

6

907587759075

Expected Value

Our raffle ticket example showed each amount had a probability associated with it

We did NOT consider the actual events but we associate numbers with the events that arose from the experiment

Expected Value

A Random Variable assigns a numerical value to all possible outcomes of a random experiment

Ex: # of heads you get when you flip a coin

twice The sum you get when you roll two dice

Expected Value

Ex. Consider tossing a coin 4 times. Let X be the number of heads. Find

and . ,3,3 XPXP 2XP

TTTTTTTHTTHTTTHH

THTTTHTHTHHTTHHH

HTTTHTTHHTHTHTHH

HHTTHHTHHHHTHHHH

S

,,,

,,,,

,,,,

,,,,

Expected Value

Soln.

1643 XP

1652 XP

TTTTTTTHTTHTTTHH

THTTTHTHTHHTTHHH

HTTTHTTHHTHTHTHH

HHTTHHTHHHHTHHHH

S

,,,

,,,,

,,,,

,,,,

16153 XP

Expected Value Note that the notation asks for the

probability that the random variable represented by X is equal to a value represented by x.

Remember that for n distinct outcomes for X,

(The sum of all probabilities equals 1).

xXP

11

n

iixXP

Expected Value

Formula for expected value (for n distinct outcomes:

n

iii xXPxXE

1

Expected value of the random variable X

Expected Value Ex. Find the expected value of X where X is

the number of heads you get from 4 tosses. Assume the probability of getting heads is 0.5.

Soln. First determine the possible outcomes. Then determine the probability of each. Next, take each value and multiply it by it’s respective probability. Finally, add these products.

Expected Value

Possible outcomes:0, 1, 2, 3, or 4 heads

Probability of each: 16

1

164

166

164

161

4

3

2

1

0

XP

XP

XP

XP

XP

TTTTTTTHTTHTTTHH

THTTTHTHTHHTTHHH

HTTTHTTHHTHTHTHH

HHTTHHTHHHHTHHHH

S

,,,

,,,,

,,,,

,,,,

Expected Value

Take each value and multiply it by it’s respective probability:

Add these products0 + 0.25 + 0.75 + 0.75 + 0.25 =

2

25.0444

75.0333

75.0222

25.0111

0000

161

164

166

164

161

XP

XP

XP

XP

XP

Expected Value

Ex. A state run monthly lottery can sell 100,000 tickets at $2 apiece. A ticket wins $1,000,000 with probability 0.0000005, $100 with probability 0.008, and $10 with probability 0.01. On average, how much can the state expect to profit from the lottery per month?

Expected Value Soln. State’s point of view:

Earn: Pay: Net:$2 $1,000,000 -$999,998$2 $100 -$98$2 $10 -$8$2 $0 $2

These are the possible values. Now find probabilities

Expected Value Soln. State’s point of view:

We get the last probability since the sum of all probabilities must add to 1.

0000005.0998,999 XP 008.098 XP 01.08 XP 9819995.02 XP

Expected Value

Soln. State’s point of view:Finally, add the products of the values and

their probabilities

60.0$

9819995.0201.08

008.0980000005.0998,999

2288

9898998,999998,999

XPXP

XPXPXE

Expected Value

Focus on the Project: X: amount of money from a loan work out Compute the expected value for typical

loan:

000,991,1$

536.0000,250$464.0000,000,4$

000,250$000,250$000,000,4$000,000,4$

Failure FailureSuccess Success

XPXP

PPXE

Expected Value Focus on the Project:

What does this tell us?

Foreclosure: $2,100,000Ave. loan work out: $1,991,000

Tentatively, we should foreclose. This doesn’t account for the specific characteristics of J. Sanders. However, this could reinforce or weaken our decision.