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At a particular carnival, there is a dice game that costs $5 to play.

At a particular carnival, there is a dice game that costs $5 to play. -If the die lands on an odd number, you lose. -If the die lands on a 2 or 4, you win $8. -If the die lands on 6, you win $14. How much could I “walk away with” for each of the possible outcomes?.

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At a particular carnival, there is a dice game that costs $5 to play.

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  1. At a particular carnival, there is a dice game that costs $5 to play. -If the die lands on an odd number, you lose.-If the die lands on a 2 or 4, you win $8.-If the die lands on 6, you win $14. How much could I “walk away with” for each of the possible outcomes?

  2. How much could I “walk away with” for each of the possible outcomes? -Lands on an odd number: How much did you pay? $5 How much money did you win/get back? - I did not get any of the money back Did you walk away with more or less $? - I walk away losing $5

  3. How much could I “walk away with” for each of the possible outcomes? -Lands on a 2 or 4: How much did you pay? $5 How much money did you win/get back? - I got back $8 Did you walk away with more or less $? - I walk away with $3 more than I started

  4. How much could I “walk away with” for each of the possible outcomes? -Lands on a 6: How much did you pay? $5 How much money did you win/get back? - I got back $13 Did you walk away with more or less $? - I walk away with $8 more than I started

  5. Did you walk away with more or less $? I walk away with $8 more than I started? The overall amount you “walk away with” (positive or negative) is called the: Net Gain

  6. There is a game at the fair where you pay $10 to flip a Coin once -If the coin lands heads up, you lose.-If the coin lands tails up, you win $19 How much could I “walk away with” for each of the possible outcomes?

  7. How much could I “walk away with” for each of the possible outcomes? -Lands heads up: How much did you pay? $10 How much money did you win/get back? - I did not get any of the money back Did you walk away with more or less $? - I walk away losing $10

  8. How much could I “walk away with” for each of the possible outcomes? -Lands tails up: How much did you pay? $10 How much money did you win/get back? - I got back $19 Did you walk away with more or less $? - I walk away with $9 more than I started

  9. What is your net gain if you lose? If you lose, the net gain = -10

  10. What is your net gain if you win? If you win, the net gain = 9

  11. Have you ever wondered…….. When playing a game, your chances May seem good, but do you think That the odds are in your favor?

  12. Anything deal with chance such Such as a casino or lottery…. What does a business have to do to In order to be successful?

  13. Therefore…. At the end of the day, the “business” Will have a positive net gain and the “players” will have an overall Negative net gain

  14. Back to our dice example….. At a particular carnival, there is a dice game that costs $5 to play. -If the die lands on an odd number, you lose.-If the die lands on a 2 or 4, you win $8.-If the die lands on 6, you win $13. Now we can create a probability Distribution with out possible Outcomes and our net gains

  15. At a particular carnival, there is a dice game that costs $5 to play. -If the die lands on an odd number, you lose.-If the die lands on a 2 or 4, you win $8.-If the die lands on 6, you win $13. What could be your possible “winnings”? Lose, win $8, win $13

  16. Lose Win $8 “Winnings” Win $13 Net Gain -5 8 3 P(X) 1/6 3/6 2/6 Now find the “mean” using net gain and P(X) Mean = -5 (3/6) + 3 (2/6) + 8 (1/6) Mean = -2.5 + 1 + 1.33 Mean = -0.2

  17. This “mean” we found using net gain and P(X) Is called: Expected Value = E(X) Therefore, each time I play the dice game I am Expected to lose $0.20 on average. Does this seem correct that I expect to lose? Yes, because that means the “business” is making $

  18. Find the expected value from our coin example There is a game at the fair where you pay $10 to flip a Coin once -If the coin lands heads up, you lose.-If the coin lands tails up, you win $19 Lose Win $8 “Winnings” Net Gain -10 9 P(X) 1/2 1/2 E(X) = -10 (1/2) + 9 (1/2) E(X) = -5 + 4.5 = -0.5

  19. Example 1: Find the expected value if tickets are sold in a raffle at $2 each. The prize is a $1000 shopping spree at a local Mall. Assume that one ticket is purchased. Lose Win “Winnings” Net Gain -2 998 1499 1500 _1__ 1500 P(X) E(X) = -2(1499/1500)+ 998(1/1500) E(X) = -1.999 + 0.665 = -1.33

  20. Example 2: Find the expected value for example #1 if two tickets Are purchased Lose Win “Winnings” Net Gain -4 996 1498 1500 _2__ 1500 P(X) E(X) = -4(1498/1500)+ 996(2/1500) E(X) = -3.995 + 1.328 = -2.67

  21. Example 3: A lottery offers one $1000 prize, one $500 prize, and Five $100 prizes. One thousand tickets are sold at $3 each. Find the expected value of one ticket. Lose Win $1000 Win $500 Win $100 “Winnings” Net Gain -3 997 497 97 993_ 1000 _1__ 1000 _1__ 1000 _5__ 1000 P(X) E(X) = -3(993/1000)+ 997(1/1000) + 497(1/1000) + 97(5/1000) E(X) = -2.979 + 0.997 + 0.497 + 0.485 = -1.00

  22. Try some on your own: One thousand tickets were sold at $1 each for four Prizes of $100, $50, $25, and $10. What is the Expected value if a person purchases two tickets? Lose Win $100 Win $50 Win $25 Win $10 “Winnings” Net Gain -2 98 48 23 8 992_ 1000 _2__ 1000 _2__ 1000 _2__ 1000 _2__ 1000 P(X) E(X) = -1.63

  23. Try some on your own: You pay $5 to draw a card from a standard deck of 52 Cards. If you pick a red card, you win nothing. If you Get a spade, you win $5. If you get a club, you win $10. If you get the ace of clubs, you win an additional $20. Find the expected value of drawing one card. Red Spade Club Ace of Clubs “Winnings” Net Gain -5 0 5 25 26 52 13 52 12 52 1_ 52 P(X) E(X) = -0.87

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