1. Expected Value MATH 102 Contemporary Math S. Rook

2. Overview Section 14.4 in the textbook: – Expected value

3. Expected Value

4. Consider playing the following game: – Flip two coins simultaneously – If both coins show Heads, you win $5 – Otherwise, you lose $2 Would you play this game? What is the probability of you winning? Losing? Given these probabilities, we can calculate how much you can expect to win or lose by playing this game multiple times Computing a prediction of data over a period of time using probability is known as expected value

5. Expected Value (Continued) In reference to gambling, expected value can be used to show why casinos win in the long run – Even though payouts may be large in games of chance, the probability of winning is slim! When you buy insurance, you are “betting” that something bad will happen to your investment while the company is “betting” nothing will happen – The probability of nothing happening to your investment is usually high – Insurance companies wish to make a profit – your premium is calculated so that the expected value of the transaction favors the company

6. Expected Value (Example) Ex 1: The table below shows average weekly profits, rounded to the nearest hundred, for a number of current franchises. If someone were to buy a franchise, what would the expected weekly profit be? Average Weekly ProfitNumber Who Earned This $1004 $2008 $30013 $40021 $5003 $6001

7. Expected Value (Example) Ex 2: The table below shows the number of fire hydrants illegally (rounded to the nearest ten) opened daily during heat waves in a certain city. How many fire hydrants should the city expect to be opened illegally per day for the upcoming heat wave? Hydrants Illegally OpenedDays 2013 3011 4015 5011 609 701

8. Expected Value (Example) Ex 3: Assume that the probability of a 25-year- old male living to age 26 is 0.98. If a $1,000 one-year term life insurance policy on a 25- year-old male costs $27.50, what is its expected value from the point of view of the company? Assume that the $27.50 need not be deducted from the $1,000 payout in the event of the male’s death.

9. Expected Value (Example) Ex 4: Suppose you play a game by rolling a single die. If the number is 5 or 6, you win $5; otherwise, you lose $3. How much can you be expected to win or lose over the long haul?

10. Expected Value (Example) Ex 5: A company estimates that it has a 60% chance of being successful in bidding on a $50,000 contract. If it costs $5,000 in consulting fees to prepare the bid, what is the expected gain or loss for the company if it decides to bid on the contract?

11. Summary After studying these slides, you should know how to do the following: – Compute the expected value Additional Practice: – See problems in Section 14.4 Next Lesson: – Organizing & Visualizing Data (Section 15.1)