Chapter 5.2Expectation

Another concept related to the mean for a probability distribution is that of expected value or expectation.

Expected value is used in various types of games of chance, in insurance, and other areas, like decision theory.

Expectation

The expected value of a discrete random variable of a probability distribution is the theoretical average of the variable.

Formula:

E(X) represents the expected value

Expectation

)()( XPXXE

One thousand tickets are sold at $1 each for a television valued at $350. What is the expected value of the gain if you purchase one ticket?

Winning Tickets

A special six-sided die is made in which 3 sides have six spots, two sides have four spots, and one side has one spot. If the die is rolled find the expected value of the number of spots that will occur.

Special Die

A financial adviser suggests that his client select one of two types of bonds in which to invest $5000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a 2.5% return and a default rate of 1%. Find the expected rate of return and decide which bond would be a better investment. (When the bond defaults, the investor loses all the investment)

Bond Investment

Chuck-a-luck This game (found in fairgrounds) is played by rolling 3 dice and betting on a number between 1 and 6. You win your bet multiplied by the number of times your chosen value appears on the three dice. For example if you bet $1 on 5 and roll 4; 5; 5 you win $2. Find the expected value of betting $1 on 4.

Chuck-A-Luck

At the Las Vegas roulette (with 38 numbers, 0,00,1,2,3, etc) you can do various bets (let's say the bet size is $1). Find the expected value of betting on each of the following

1. Bet on red (or black) and a successful bet pays you $1.

2. Bet on a number and a successful bet pays you $35.

3. Bet on the first (or the second, the third) dozen of numbers a successful bet pays you $2.

Las Vegas Roulette

Applying the concepts, Pg. 274 #1-7

Try it!