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Expected Value

- Chapter 20

Expected value

- expected value a weighted average we average

the outcomes but give more weight to those

outcomes that occur more often (higher

probability) - expl expected number of vehicles per household

data from sample of households

- You cannot just take an average of the six

possible outcomes (number of vehicles) because

that would give them all equal weight. - You need to take into consideration that the

probability that a household has 2 vehicles is

more than the probability that a household has 4

vehicles, etc.

- Let X represent the number of vehicles.
- Expected value of X denoted E(X)
- 0(.04) 1(.25) 2(.45) 3(.18) 4(.06)

5(.02) - 2.03
- So the expected number of vehicles in a household

is 2.03. This is a weighted average of all the

households sampled. If we were to select a

household randomly, wed expect them to have

about 2.03 vehicles.

- In general, let a1, a2, a3, , ak be possible

outcomes and p1, p2, p3, , pk be their

probabilities. - Then E(X) a1 p1 a2 p2 ak pk
- expl Lottery
- A lottery ticket costs 1. You pick a three digit

number. If your number is chosen, you win 500. - We will find the expected value of this lottery

ticket (from the consumers perspective). This

will tell us, on average, how much each consumer

can expect to make or lose. - First, find the probability of winning and the

probability of losing as well as the actual

winnings.

If I spend 1 for the ticket and win nothing, my

winnings are -1. If I spend 1 and win 500, my

winnings are 499.

There are 1000 3-digit numbers (total

possibilities). The winner will be 1 of those (a

success).

Worksheet and Homework

- E(X) -1(.999) 499(.001)
- -.50
- So the consumer can expect to lose 50 cents, on

average, each time he or she plays.

Expected value of lottery ticket discusses the

expected value of a Illinois Lotto lottery

ticket. We find expected value a little

differently than the book does. Be careful here.

Chapter 20 homework 1, 3, 6, 11, 16

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