Probability

Module 8: Probability Binomial Distribution

Expected Value

Lesson

Expected value is the sum of the probability of each possible outcome of an ev ent multiplied by the corresponding v alue of that outcome. In other words, it is the sum of the products of the ev ents' probabilities and their v alues. The expected v alue is a weighted av erage of the possible outcomes. It is a weighted av erage because each outcome is weighted by its probability. It can represent the amount that is predicted to be gained or lost af ter play ing a game or in a real-lif e situation. The expected v alue may not be an actual outcome, as it represents an av erage v alue, which could lie between two actual outcome v alues.

The f ormula f or expected v alue is:

E = p1x1 + p2x2 + p3x3 +...+ pnxn , where p is the probability of each outcome and x is the v alue of each outcome.

The f ormula is sometimes written as: . The Greek letter Sigma (Σ) alway s means to take the sum. The f ormula say s to take the sum of ALL the products

of the ev ents and their probabilities. Expected v alue is a weighted sum.

Examples

Example 1: Suppose y ou are play ing a game with a number cube that has its six f aces numbered 1, 1, 1, 1, 2, and 3. What would be the expected v alue f or rolling the cube?

Solution:

Determine the probability of each outcome, and organize y our results into a table:

Outcome v alue, x 1 2 3

Probability, P P(1) = 4 f aces with number 1 out of 6 f aces

= 4/6 = 2/3

P(2) = 1 f ace with number 2 out of 6 f aces

= 1/6

P(3) = 1 f ace with number 3 out of 6 f aces

= 1/6

Note: The sum of P = 2/3 + 1/6 + 1/6 = 4/6 + 1/6 + 1/6 = 6/6 = 1. If y our probablities do not sum to one, the expected v alue will not be accurate.

The expected v alue is the sum of each outcome v alue multiplied by its probability : E = p1x1 + p2x2 + p3x3 (2/3)(1) + (1/6)(2) + (1/6)(3) = 2/3 + 2/3 + 1/2 = 1.5

Example 2: You hav e been approached by some of y our f riends that want y ou to participate in a "Game of Chance." They explain the rules of the game as f ollows: You spin a spinner that has 6 equal size sectors. Two sectors are blue, and the rest are red. If y ou spin blue, y ou get two points and if y ou spin red, y ou lose one point. How do y ou know if the game is f air?

Solution: Find the expected v alue. A f air game is a game in which the expected v alue of the game is 0. In other words, each play er has an equal chance of winning. You need to look at the probability and outcome v alue f rom y our point of v iew, the v alues may be dif f erent f or y our opponents.

Outcome v alue, x Blue: + 2 points Red: - 1 point

Probability, P P(blue) = 2 blue sectors out of 6 total sections = 2/6 P(red) = 4 red sectors out of 6 total sections = 4/6

Note: The sum of P = 2/6 + 6/6 = 6/6 = 1. If y our probablities do not sum to one, the expected v alue will not be accurate.

E = p1x1 + p2x2 = (2/6)(2) + (4/6)(-1) = 2/3 + (-2/3+ = 0

Since the expected v alue is zero, the game is f air.

Example 3: Suppose that in one y ear, there were 4200 house f ires in approximately 500,000 homes and that the av erage y early premium f or f ire insurance f or homeowners was $225. What was the expected v alue of insurance cov erage f or a person with homeowners f ire insurance if the av erage claim paid by the insurance company was $10,000?

Solution: This problem requires a bit of work bef ore y ou organize y our data. The probability of hav ing a f ire is 4200/500,000, apx 0.0084.

P(no f ire) = 1 - 0.0084 = 0.9916. If there is a f ire, outcome is the pay out less the premium, 10,000 - 225 = 9,775. If there isn't a f ire, the outcome is the premium paid, -225.

Outcome v alue, x Fire: 9775 No f ire: -225

Probability, P P(f ire) = 0.0084 P(no f ire) = 0.9916

The sum of P = .0084 + .9916 = 1. If y our probablities do not sum to one, the expected v alue will not be accurate.

E = p1x1 + p2x2 = (0.0084)(9775) + (0.9916)(-225) = - $141. You can expect to lose $141 per y ear.

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Example 4: You are taking a test where ev ery correct answer scores 4 points and ev ery incorrect answer scores -1 point. There are 4 possible choices f or each question. If y ou don't know the answer, should y ou guess?

Solution: Determine the expected v alue. If it is positiv e, it is to y our adv antage to guess. If it is negativ e, it is not to y our adv antage to guess. If it is 0, it is neither to y our adv antage nor disadv antage to guess.

Outcome v alue, x Guess correct: + 4 points Guess incorrect: -1 point

Probability, P P(correct) = 1 correct answer out of 4 choices = 1/4 P(incorrect) = 3 incorrect answers out of 4 choices = 3/4

The sum of P = 1/4 + 3/4 = 4/4 = 1. If y our probablities do not sum to one, the expected v alue will not be accurate.

E = p1x1 + p2x2 = (1/4)(4) + (3/4)(-1) = 1 + (-3/4) = 1/4

Since the expected v alue is positiv e, it is to y our adv antage to guess.

Example 5:

The high school band sales raf f le tickets f or $10 each. They will be selling a total of 500 tickets and the winner will receiv e $1000. What could y ou expect to win or what is y our expected v alue?

Solution: Determine the expected v alue. If it is positiv e, y ou will win money by partipating in the raf f le. If it is negativ e, y ou will expect to lose money by participating in the raf f le.

Outcome v alue (x) Win: $1000 - $10 = $990 Lose: - $10

Probability, (P) P (win) - 1/500 P(Lose) = 499/500

The sum of P = 1/500 + 499/500 = 500/500 = 1. If y our probablities do not sum to one, the expected v alue will not be accurate.

E = p1x1 + p2x2 = (1/500)($990) + (-10)(499/500) = 1.98 - 9.98 = -8

Since the expected v alue is negativ e, y ou can expect to lose money play ing this raf f le.

Example 6:

At a raf f le, 500 tickets are sold at $1 each f or three prizes of $100, $50, and $10. What is the expected v alue of y our net gain if y ou buy a ticket?

Solution:

Construct a probability distribution f or the possible net gains. Then f ind the expected v alue. The net gain f or each prize is the v alue of the prize minus the cost of the tickets purchased.

Gain (x) $100 - 1 = $99 $50 - 1 = $49 $10 - 1 = $9 $0 - 1 = - $1

Probability, P(x)

E(x) = (99 ∙ 0.002) + (49 ∙ 0.002) + (9 ∙ 0.002) + (-1 ∙ 0.994) = - .68

This expected v alue means that the av erage loss f or someone purchasing a ticket is $0.68.

Example 7:

An open-air restaurant on the beach loses $90,000 per season when the weather is rainier than usual and makes $450,000 when the weather is normal. If the probability of hav ing weather that is rainier than normal this season is 20%, f ind the restaurants expected prof it?

Solution:

If the weather is rainier than normal 20% of the season, then the weather is normal 80% of the season.

E(x) = .2(-90,000) + .8(450,000) = $342,000

The restaurant is expected to prof it $342,000.

Note that f or each example, the sum of the probabilities in the table is equal to 1. This is the case f or all expected v alue problems. Watch f or rounding errors, do NOT round each step of the problem. Key it into y our calculator exactly as it appears, using parenthesis. Round only at the end.

Example: (1/3500)(975) + (1/3500)(475) + (1/3500)(75) + (1/3500)(-25) = -24.52 If each section is computed and rounded separately, y ou will get 0.28 + 0.30 + 0.02 + (-24.97) = -24.38

Self-Check

Self-Check Problem 1

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Consider a game in which two play ers each choose an integer f rom 1 to 3. If the sum of

the 2 integers is ev en, then play er A scores 4 points and play er B loses 2 points. If the

sum is of f , then play er B scores 4 points and play er A loses 2 points.

a. Find the expected v alue f or Play er A.

b. Is the game f air?

a. 1.33; f air

b. 1.33; not f air

c. -1.33; f air

d. -1.33; not f air

C heck A nswer

Self-Check Problem 2

A water park makes $350,000 when the weather is normal and loses $80,000 per season

when ther are more bad weather day s than normal. If the probability of hav ing more bad

weather day s than normal this season is 35%, f ind the park's expected prof it.

C heck A nswer

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