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Section 8.2: Expected Values

Section 8.2: Expected Values. What is an expected value ?. An expected value is found by multiplying each outcome (“a” values) by its probability (“p” values) and then adding the products of each of these. In symbols, it looks like the following:.

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Section 8.2: Expected Values

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  1. Section 8.2: Expected Values

  2. What is an expected value? • An expected valueis found by multiplying each outcome (“a” values) by its probability (“p” values) and then adding the products of each of these. • In symbols, it looks like the following:

  3. What is the expected value for a lottery ticket? • Example: The “Straight” from the Pick 3 game of the Tri-State Daily Numbers is offered by New Hampshire, Maine and Vermont. The state chooses a three-digit winning number at random and pays you $500 if your number is chosen. Because there are 1000 three-digit numbers, you have probability 1/1000 of winning. Here is the probability model:

  4. Using Expected Values… • 500(0.001) + 0(0.999) = $0.50 or 50 cents. • So, in the long run, the state pays out half of the money bet, and keeps the other half.

  5. Another Example… • “Straight-Box” is another way of playing the Pick 3 game. In this game, you have 2 ways to win: You win $292 if you exactly match the winning number, and you win $42 if your number has the same digits, in any order. • The probability model for this is: • Remember, there are 6 ways of arranging 3 numbers (ex., 123 can also be arranged as 132, 312, 321, 213, and 231.)

  6. So the Expected Value is… • ($0)(0.994) + ($42)(0.005) + ($292)(0.001) • =$0.502, which is slightly more than the Straight bet, so the state in the long run, pays out more in this type of lottery than it does in the Straight bet (which was $0.50).

  7. And another example… • What is the average number of motor vehicles in American households? The Census Bureau tells us that the distribution of vehicles per household (as of 1997) is as follows: • Expected value = ? • 2.03…the average number of motor vehicles per household is 2.03.

  8. Something to remember… • The law of large numbers says the actual average outcome of many trials gets closer to the expected value as more trials are made, but it doesn’t say how many trials are needed. • The more variable the outcomes, the more trials are needed.

  9. The next 3 questions… • If your group gets it right, your group gets 3 points; otherwise your group loses 3 points. • The group with the most points at the end of 3 questions gets extra credit (in the form of how many points they have earned). • In the event of a tie, a tiebreaker question will be issued.

  10. Question #1 • Suppose that in a certain game, a player rolls an ordinary six-sided number cube. If the cube lands on a 6, the player is paid $4. If the cube lands on any other number, the player loses $1. What is the player’s mathematical expectation?

  11. Question #2 • An American roulette wheel has 38 numbers on it. Suppose a person bets $1 on a specific number. The wheel is spun, and if the number picked comes up, the person wins $35; otherwise, the person loses his or her $1 bet. What is the mathematical expectation?

  12. Question #3 • Sometimes mathematical expectation is positive. Suppose a manufacturer makes a profit of $3 on every item sold but loses $12 on each defective one. If 1% of the items are defective, what is the mathematical expectation (profit)?

  13. Question #4 • In the insurance industry, actuaries calculate how much to charge so that the company is competitive and can still make a profit. Suppose that for people of a certain age, the probability of dying the coming year is 1/200 and a customer of that age buys a $1000 insurance policy for $7. What is the mathematical expectation of the customer?

  14. Homework • Page 468..480 • #8.19-8.21, 8.24, 8.28

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