Section 11-5

1 / 18

# Section 11-5 - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## Section 11-5

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Section 11-5
• Expected Value
Expected Value
• Expected Value
• Games and Gambling
• Investments
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 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 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

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 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

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 Winnings

Solution

The information for the game is displayed below.

Expected value: E(x) = –\$6/6 = –\$1.00

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

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

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

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 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

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

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 Revenue

Solution

The expected revenue from sales can be found below.

Expected revenue: E(x) = \$9050

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.