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Section 11-5. Expected Value. Expected Value. Expected Value Games and Gambling Investments Business and Insurance. Expected Value.

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section 11 5
Section 11-5
  • Expected Value
expected value
Expected Value
  • Expected Value
  • Games and Gambling
  • Investments
  • Business and Insurance
expected value1
Expected Value

Children in third grade were surveyed and told to pick the number of hours that they play electronic games each day. The probability distribution is given below.

expected value2
Expected Value

Compute a “weighted average” by multiplying each possible time value by its probability and then adding the products.

1.1 hours is the expected value (or the mathematical expectation) of the quantity of time spent playing electronic games.

expected value3
Expected Value

If a random variable x can have any of the values x1, x2 , x3 ,…, xn, and the corresponding probabilities of these values occurring are

P(x1), P(x2), P(x3), …, P(xn), then the expected value of xis given by

example finding expected value
Example: Finding Expected Value

Find the expected number of boys for a three-child family. Assume girls and boys are equally likely.

Solution

S = {ggg, ggb, gbg, bgg, gbb, bgb, bbg, bbb}

The probability distribution is on the right.

example finding expected value1
Example: Finding Expected Value

Solution(continued)

The expected value is the sum of the third column:

So the expected number of boys is 1.5.

example finding expected winnings
Example: Finding Expected Winnings

A player pays $3 to play the following game: He rolls a die and receives $7 if he tosses a 6 and $1 for anything else. Find the player’s expected net winnings for the game.

example finding expected winnings1
Example: Finding Expected Winnings

Solution

The information for the game is displayed below.

Expected value: E(x) = –$6/6 = –$1.00

games and gambling
Games and Gambling

A game in which the expected net winnings are zero is called a fair game. A game with negative expected winnings is unfair against the player. A game with positive expected net winnings is unfair in favor of the player.

example finding the cost for a fair game
Example: Finding the Cost for a Fair Game

What should the game in the previous example cost so that it is a fair game?

Solution

Because the cost of $3 resulted in a net loss of $1, we can conclude that the $3 cost was $1 too high. A fair cost to play the game would be $3 – $1 = $2.

investments
Investments

Expected value can be a useful tool for evaluating investment opportunities.

example expected investment profits
Example: Expected Investment Profits

Mark is going to invest in the stock of one of the two companies below. Based on his research, a $6000 investment could give the following returns.

example expected investment profits1
Example: Expected Investment Profits

Find the expected profit (or loss) for each of the two stocks.

Solution

ABC: –$400(.2) + $800(.5) + $1300(.3) = $710

PDQ: $600(.8) + $1000(.2) = $680

business and insurance
Business and Insurance

Expected value can be used to help make decisions in various areas of business, including insurance.

example expected lumber revenue
Example: Expected Lumber Revenue

A lumber wholesaler is planning on purchasing a load of lumber. He calculates that the probabilities of reselling the load for $9500, $9000, or $8500 are .25, .60, and .15, respectfully. In order to ensure an expected profit of at least $2500, how much can he afford to pay for the load?

example expected lumber revenue1
Example: Expected Lumber Revenue

Solution

The expected revenue from sales can be found below.

Expected revenue: E(x) = $9050

example expected lumber revenue2
Example: Expected Lumber Revenue

Solution(continued)

profit = revenue – cost or cost = profit – revenue

To have an expected profit of $2500, he can pay up to $9050 – $2500 = $6550.