Math 103 - Cooley Statistics for Teachers OCC

Math 103 - Cooley Statistics for Teachers OCC

Activity #25 – Expected Value

Expected Value

Suppose that the outcomes of an experiment are real numbers (values) called and suppose that the outcomes have probabilities , respectively. The expected value, E, of the

experiment is the sum

J Example:

A bag contains: 2 $1 bills, 1 $5 bill, 1 $10 bill, 1 $20 bill, 1 $50 bill and 1 $100 bill. A student reaches in

the bag and selects a bill from the seven bills at random. Here we assume that the bills are indistinguishable

to the student in terms of touch. What is the expected value of the event which consists of reaching in the bag and selecting one bill at random?

à Solution:

Let the values be the denomination of each bill.

Let the values be the probabilities of each bill drawn at random.

So this means on the average, one would win $26.71 each time they play.

** So if someone charged you $20.00 to play each time, you would win a net of $6.71. This is considered

an unfair game, yet the advantage goes to the player. ($26.71 - $20.00 = +$6.71)

** So if someone charged you $30.00 to play each time, you would lose a net of $3.29. This is considered

an unfair game, and the advantage goes to the house. ($30.00 - $26.71 = -$3.29)

** So if someone charged you $26.71 to play each time, you would break even. This is considered a fair

game, and there is no advantage to either side. ($26.71 - $26.71 = $0.00)

J Exercises:

1) A bag contains: 5 $1 bills, 1 $5 bills, 5 $10 bills, and 1 $20 bill. You reach in the bag and select a bill at random. What is the expected value?

2) According to a publisher's records, 20% of the books published break even, 30% lose $1000, 25% lose $10,000, and 25% earn $20,000. When a book is published, what is the expected income for the book?

3) A bag contains: 2 $1 bills, 3 $2 bills, and 1 $50 bill. You reach in the bag and select a bill at random. It costs $20 to play. What is the expected value? Is this a fair game? Would you play?

4) Roll a pair of standard dice. If a sum of 6, 7, or 8 is rolled, you lose $2.00. If a 4, 5, 9, or 10 is rolled, you lose $1.00. If a 2, 3, or 11 is rolled, you win $5.00. If a 12 is rolled, you win $20. Without doing any math, who would you think has the advantage, the house or you? Show who does have the advantage, if any, by finding the expected value.

J Exercise:

5) Hosted by Howie Mandel, "Deal or No Deal" is an exhilarating hit game show where contestants play and deal for a top prize of $1 million in a high-energy contest of nerves, instincts and raw intuition. Each night, the game of odds and chance unfolds when a contestant is confronted with 26 sealed briefcases full of varying amounts of cash - ranging from a measly penny to $1 million. Without knowing the amount in each briefcase, the contestant picks one -- his to keep, if he chooses - until its unsealing at game's end. The risk element kicks in when the player must then instinctively eliminate the remaining 25 cases - which are opened and the amount of cash inside revealed. The pressure mounts as in each round, after a pre-determined number of cases are opened, the participant is tempted by a mysterious entity known only as "the Banker" to accept an offer of cash in exchange for what might be contained in the contestant's chosen briefcase - prompting Mandel to ask the all-important question - Deal or No Deal? As each case is opened, the likelihood of the player having a valuable cash amount in his or her own case decreases or increases. Viewers will see if, truly, fortune favors the bold. The contestant knows that as long as the larger cash prizes haven't been opened, the Banker's deals will only get higher. And if the conflicted contestant accidentally opens a case with a bigger cash value - the Banker's offer could suddenly evaporate.

The following graphic below shows the 26 different denominations.

If you were to select a suitcase at random, what is the expected value?

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