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Warm-up Use the table to answer the questions. 1.What is the probability that someone wearing their seatbelt was going >15 mph over speed limit? 2.What is the probability that someone was going 10-15 mph OR was not wearing a seatbelt? Number of mph over speed limit 10-15>15Total Wearing Seatbelt Yes65145210 No157590 Total80220300

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Expected Value Essential Question - What is expected value used for in real-life?

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What it is and what it is used for A weighted average The expected results of an experiment in the long run. Used in business to predict future profits Used in insurance to determine how much a persons insurance rate Used in games such as the lottery, slot machines, roulette to determine expected winnings (or losses)

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How you find it Multiply each probability by amount you will win and then add all together Used in business to predict future profits Used in insurance to determine how much a persons insurance rate Used in games such as the lottery, slot machines, roulette to determine expected winnings (or losses)

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Example I will give you $1 if you roll an even number on a die and you give me $1 if you roll an odd number. Who would win money in the long run? (prob of even)($1)+(prob of odd)(-$1) If the expected value is 0, the game is called FAIR

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Interpreting Expected Value If you get a ZERO expected value, you expect to BREAK EVEN in the long run If you get a POSITIVE expected value, you expect to WIN in the long run If you get a NEGATIVE expected value, you expect to LOSE in the long run The value you get for expected value will probably NOT be one of the winnings of the game

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Example 2 If you roll a 1, I will give you $4 and if you dont roll a 1, you give me $1. What is the expected value? Does this mean you will win or lose money? (prob of 1)($4) + (prob of NOT 1)(-$1) You will lose money over time

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Example 3 Suppose it costs $5 to spin the spinner and you win the amount you spin. What is your expected value? Should you play? p(2)($2-$5)+p(10)($10-$5)+p(1)($1-$5) You should not play if you want to win money

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Example 4 You are taking a multiple choice test that has 4 possible answers for each question. You get 3 points for each correct answer and lose 1 point for each incorrect answer, and do not gain or lose any points for answers left blank. If you do not know the answer to a question should you guess an answer to a question you dont know? Hint:1. Find the probability of each outcome. 2. Find the expected value of guessing the answer

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Answer 1.Each question has 4 possible answers; only 1 is correct. Guessing correctly is ¼ and guessing incorrectly is ¾ 2. Multiply the points gained or lost by the correspoding probability. 3 for correct, -1 for incorrect so…. E = 3(1/4) + (-1) (3/4) = 0 ¾ - ¾ = 0 so what is your answer? Is it advantageous to guess if you dont know the answer???

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Can we make money? At a roulette wheel there are 2 zeroes and 36 non zero numbers (18 red and 18 black) to bet on. If I bet $1 on red what is the expected value of my bet? How about after 10 of the same bets? How much can I be expected to win or lose?

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Did I win? My chance of winning $1 is 18/38 and my chance of losing $1 (or winning -$1) is 20/38. My expected value, E = (18/38) ( 1 ) + (20/38 )( -1) = 18/38 - 20/38 = -2/38 = -5.26%. I can expect to lose.0526 every spin of the wheel After 10 bets I will lose 52.6 = 53 cents. How does the casino make money if one person loses an average of 53 cents every 10 bets of $1?

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Spinner What is the expected value of the spinner? $800 $200 $500 $400 $700 $100 $300 $600

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Tables xP 3.25 4.30 13.10 2.35 Use your formula and calculate (.25) 3 + (.30) 4 + (.10) 13 + (.35) 2 =.75 + 1.20 + 1.30 +.70 = 3.95

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Tables Find the expected value of the following event. 1 173.8 30 6.85 xP 5.25 8.30 11.10 6.35

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Classwork 1.In a group with NO MORE than 4 people, you will calculate the expected value of a single one dollar scratch off lottery ticket. Show the calculations you did to get the answer, even if you used a calculator. 2.If you purchased 1000 of these tickets, what would your net loss be? 3.Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the your lottery ticket game and split your earnings? Why or why not? (In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?) 4.How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

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WIN PRIZES IN GAMEProbability:P x V $1216,366 $2120,870 $415,116 $612,581 $102,545 $501,673 $100527 $50048 -$19,642,262 There were 10,000,000 Fast 50s scratch off lottery tickets sold for $1 each. Calculate the Expected Value of the ticket. E = p 1 x 1 + p 2 x 2 + p 3 x 3 +... + p n x n

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1.If you purchased 1000 of these tickets, what would your net win/ loss be? 2.Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the your lottery ticket game and split your earnings? Why or why not? (In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?) 3.How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

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WIN (Outcome) PRIZES IN GAMEProbabilityP x V $5167,449 $1031,405 $2520,929 $502,896 $100428 $500217 $1,00080 $50,0003 -$59,776,593 There were 10,000,000 Golden Goose scratch off lottery tickets sold for $5 each. Calculate the Expected Value of the ticket. E = p 1 x 1 + p 2 x 2 + p 3 x 3 +... + p n x n

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1.If you purchased 1000 of these tickets, what would your net win/ loss be? 2.Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the your lottery ticket game and split your earnings? Why or why not? (In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?) 3.How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

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WIN (Outcome) PRIZES IN GAMEProbability:P x V $578,323 $1015,698 $205,227 $501,301 $100103 $1,00016 $50,0002 -$5899,330 There were 1,000,000 Hot Hand scratch off lottery tickets sold for $5 each. Calculate the Expected Value of the ticket. E = p 1 x 1 + p 2 x 2 + p 3 x 3 +... + p n x n

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1.If you purchased 1000 of these tickets, what would your net win/ loss be? 2.Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the your lottery ticket game and split your earnings? Why or why not? (In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?) 3.How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

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WIN (Outcome) PRIZES IN GAMEProbability:P x V $20198,405 $3066,138 $5025,666 $1004,529 $200616 $500203 $1,00015 $10,00012 $20,0004 $100,0003 $1,000,0004 -$20700,264 There were 1,000,000 $1 Million Cash Spectacular scratch off lottery tickets sold for $20 each. Calculate the Expected Value of the ticket. E = p 1 x 1 + p 2 x 2 + p 3 x 3 +... + p n x n

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1.If you purchased 1000 of these tickets, what would your net win/ loss be? 2.Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the your lottery ticket game and split your earnings? Why or why not? (In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?) 3.How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

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WIN (Outcome) PRIZES IN GAMEProbability:P x V $1201,367 $2120,779 $430,173 $515,109 $107,558 $207,551 $50625 $100122 $1,00035 -$1616,681 There were 1,000,000 Baseball 10 scratch off lottery tickets sold for $1 each. Calculate the Expected Value of the ticket. E = p 1 x 1 + p 2 x 2 + p 3 x 3 +... + p n x n

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1.If you purchased 1000 of these tickets, what would your net win/ loss be? 2.Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the your lottery ticket game and split your earnings? Why or why not? (In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?) 3.How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

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WIN (Outcome) PRIZES IN GAMEProbability:P x V $1210,344 $2120,194 $415,030 $415,018 $610,004 $102,506 $202,508 $501,567 $100762 -$1637,085 There were 1,000,000 100s of 100s scratch off lottery tickets sold for $1 each. Calculate the Expected Value of the ticket. E = p 1 x 1 + p 2 x 2 + p 3 x 3 +... + p n x n

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1.If you purchased 1000 of these tickets, what would your net win/ loss be? 2.Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the your lottery ticket game and split your earnings? Why or why not? (In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?) 3.How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

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WIN (Outcome) PRIZES IN GAMEProbability:P x V $1221,526 $2120,784 $415,118 $510,066 $102,537 $202,539 $501,584 $100749 $20044 $2,0005 -$1625,048 There were 1,000,000 Money See Money Do scratch off lottery tickets sold for $1 each. Calculate the Expected Value of the ticket. E = p 1 x 1 + p 2 x 2 + p 3 x 3 +... + p n x n

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1.If you purchased 1000 of these tickets, what would your net win/ loss be? 2.Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the your lottery ticket game and split your earnings? Why or why not? (In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?) 3.How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

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WIN (Outcome) PRIZES IN GAMEProbability:P x V $2108,983 $431,881 $533,520 $1015,103 $206,710 $50417 $100106 $10,0003 -$2803,277 There were 1,000,000 Vacation Cash scratch off lottery tickets sold for $2 each. Calculate the Expected Value of the ticket. E = p 1 x 1 + p 2 x 2 + p 3 x 3 +... + p n x n

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1.If you purchased 1000 of these tickets, what would your net win/ loss be? 2.Would it help your expected winnings if you and 9 other people bought a total of 100 tickets in the your lottery ticket game and split your earnings? Why or why not? (In other words, would you win more by pooling your resources with 9 others rather than buying only 10 tickets yourself?) 3.How does computing expected value of these tickets help a person to maintain a sensible perspective on purchasing lottery scratch game tickets?

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Homework pg. 357 #1-9 **Remember to check the ODDS in the back of the book!***

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