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Suppose I have two fair dice.

Suppose I have two fair dice. Player one gets 2 points if the sum is odd. Player two gets 4 points if the product is odd. Is this game fair?. Agenda. Review finding probability Determine expected value Is this game fair--1 player? 2 players? Fundamental Counting Principle

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Suppose I have two fair dice.

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  1. Suppose I have two fair dice. Player one gets 2 points if the sum is odd. Player two gets 4 points if the product is odd. Is this game fair?

  2. Agenda • Review finding probability • Determine expected value • Is this game fair--1 player? 2 players? • Fundamental Counting Principle • Combinations vs. Permutations

  3. Expected value • Expected value is used to determine winnings. It is related to weighted averages and probability. • Think of this one: If I flip a coin and get a head, I win $0.50. If I get a tail, I win nothing. If I flip this coin twice, what do you think I should expect to walk away with? • If I flip 4 times, what will I expect to win? • If I flip 100 times, … ? • n times…?

  4. Expected value • In general, I consider each event that is possible in my experiment. Each event has it’s own consequence (win or lose money, for example). And each event has a probability associated with it. • P(E1)•X1 + P(E2)•X2 + ••• + P(En)•Xn

  5. Here are three easy examples… • Roll a 6-sided die. If you roll a “3”, then you win $5.00. If you don’t roll a “3”, then you have to pay $1.00. • P(3) = 1/6 P(not 3) = 5/6 • P(3) • (5) + P(not 3) • (-1) = • Expected Value • (1/6)•(5) + (5/6)(-1) = 5/6 - 5/6 = 0. • If the expected value is 0, we say the game is fair.

  6. Here are three easy examples… • Roll another die. If you roll a 3 or a 5, you get a quarter. If you roll a 1, you get a dollar. If you roll an even number, you pay 50¢. P(3 or 5) = 1/3, P(1) = 1/6, P(even) = 1/2 Expected value (1/3)•(.25) + (1/6)•(1) +(1/2)•(-.50) = .0833 + .1667 -.25 = 0. Another fair game.

  7. Here are three easy examples… • Is this grading system fair? There are four choices on a multiple-choice question. If you get the right answer, you earn a point. If you get the wrong answer, you lose a point. • P(right answer) P(wrong answer) • Expected Value

  8. R R W B R Y R Y B B W W Here’s a harder one… • Suppose I spin the spinner. • Here are the rules.If I spin blue or white, I geta quarter. If I spin red,I get a nickel. If I spinyellow, I have to pay 1 dollar. • BLUE + WHITE + RED + YELLOW = 3/12 • .25 + 3/12 • .25 + 4/12 • .05 + 2/12 • (-1) = .0625 + .0625 + .0167 + (-.1667) = -.025 or -2.5¢

  9. One event • On a certain die, there are 3 fours, 2 fives, and 1 six. • P(rolling an odd) = • P(rolling a number less than 6) = • P(rolling a 6) = • P(not rolling a 6) = • P(rolling a 2) = • Name two events that are complementary. • Name two events that are disjoint.

  10. Two events • I have 6 blue marbles and 4 red marbles in a bag. If I do not replace the marbles, … • P(blue) = • P(red) = • P(blue, blue) = • P(red, blue) = • P(blue, red) = • Is this an example of independent or dependent events?

  11. Two events • There are 8 girls and 7 boys in my class, who want to be line leader or lunch helper, … • P(G: LL, B: LH) = • P(G: LL, G: LH) = • P(B: LL, B: LH) = • Is this an example of dependent or independent events?

  12. Watch the wording… • Suppose I flip a coin. • P(H) = • P(T) = • P(H or T) = • P(H and T) =

  13. True/False • Suppose you have a true/false section on tomorrow’s exam. If there are 4 questions,… • Make a list of all possibilities (tree diagram or organized list). • P(all 4 are true) = • P(all 4 are false) = • P(two are true and two are false) = • Is this an example of independent or dependent events?

  14. Shortcut! • If drawing a tree diagram takes too long, consider this shortcut. • Now, what do we do with these numbers? 1st Q 2nd Q 3rd Q 4th Q

  15. Fundamental Counting Principle • So, for the true/false scenario, it would be:true or false for each question.2 • 2 • 2 • 2 = 16 possible outcomes of the true/false answers. Of course, only one of these 16 is the correct outcome. • So, if you guess, you will have a 1/16 chance of getting a perfect score. • Or, your odds for getting a perfect score are 1 : 15.

  16. Fundamental Counting Principle • Suppose you have 5 multiple-choice problems tomorrow, each with 4 choices. How many different ways can you answer these problems? • 4 • 4 • 4 • 4 • 4 = 1024

  17. Fundamental Counting Principle • Now, suppose the question is matching: there are 6 questions and 10 possible choices. Now, how many ways can you match? • 10 • 9 • 8 • 7 • 6 • 5 = 151,200

  18. How are true/false and multiple choice questions different from matching questions?

  19. For dependent events, … • Permutations vs. Combinations • In a permutation, the order matters. In a combination, the order does not matter. • I have 18 cans of soda: 3 diet pepsi, 4 diet coke, 5 pepsi, and 6 sprite. • Permutation or combination? • I pick 4 cans of soda randomly. • I give 4 friends each one can of soda, randomly.

  20. Examples • I have 12 flowers, and I put 6 in a vase. • I have 12 students, and I put 6 in a line. • I have 12 identical math books, and I put 6 on a shelf. • I have 12 different math books, and I put 6 on a shelf. • I have 12 more BINGO numbers to call, and I call 6 more--then someone wins.

  21. Permutations and Combinations • In a permutation, because order matters, there are more outcomes to be considered than in combinations. • For example: if we have four students (A, B, C, D), how many groups of 3 can we choose? • In a permutation, the group ABC is different than the group CAB. In a combination, the group ABC is the same as the group CAB.

  22. Combinations: don’t count duplicates • So, how do I get rid of the duplicates? • Let’s think. • If I have two objects, A and B…then my groups are AB and BA, or 2 groups. • If I have three objects, A, B, and C…then my groups are ABC, BAC, ACB, BCA, CAB, CBA, or 6 groups.

  23. If I have three objects, A, B, and C…then my groups are ABC, ACB, BAC, BCA, CAB, CBA, or 6 groups. • If I have 4 objects A, B, C, and D… • Build from ABC:DABC, ADBC, ABDC, ABCD • Now build from ACB: • DACB, ADCB, ACDB, ACBD • Keep going… How many possible?

  24. Factorial • So, for 5 objects A, B, C, D, E, … • It will be 5 • 4 • 3 • 2 • 1. • We call this 5 factorial, and write it 5! • See how this is related to the Fundamental Counting Principle?So, if there are 5 objects to put in a row, then there is 1 combination, but 120 permutations.

  25. Two more practice problems • Suppose I have 16 kids on my team, and I have to make up a starting line-up of 9 kids. • Permutation or combination: kids in the field (don’t consider the position). Solve. • Permutation or combination: kids batting order. Solve.

  26. Kids in the field--the order of which kid goes on the field first does not matter. We just want a list of 9 kids from 16. • 16 • 15 • 14 • 13 • 12 • 11 • 10 • 9 • 8 • Divide by 9! (to get rid of duplicates). • Write it this way: 16 • 15 • 14 • 13 • 12 • 11 • 10 • 9 • 8 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1

  27. Combinations: 11,440 • Permutations: 4,151,347,200 • Since the batting order does matter, this is an example of a permutation.

  28. Another example • My bag of M&Ms has 4 blue, 3 green, 2 yellow, 4 red, and 8 browns--no orange. • P(1st M&M is red) • P(1st M&M is not brown) • P(red, yellow) • P(red, red) • P(I eat the first 5 M&Ms in this order: blue, blue, green, yellow, red) • P(I gobble a handful of 2 blues, a green, a yellow, and a red)

  29. Homework • Due on Tuesday: do all, turn in the bold. • Section 7.4 p. 488 #2, 3, 7, 8, 12, 13, 15 • Read section 8.1

  30. Deal or no Deal • You are a contestant on Deal or No Deal. There are four amounts showing: $5, $50, $1000, and $200,000. The banker offers $50,000. • Should you take the deal? Explain. • How did the banker come up with $50,000 as an offer?

  31. A few practice problems • A drawer contains 6 red socks and 3 blue socks.P(pull 2, get a match)P(pull 3, get 2 of a kind)P(pull 4, all 4 same color)

  32. How many different license plates are possible with 2 letters and 3 numbers? (omit letters I, O, Q) Is this an example of independent or dependent events? Explain.

  33. Review Permutations and Combinations • I have 10 different flavored popsicles, and I give one to Brendan each day for a week (7 days). • How many ways can I do this? • 10 • 9 • 8 • 7 • 6 • 5 • 4 • This is a permutation.

  34. Review permutations and combinations • Janine’s boss has allowed her to have a flexible schedule where she can work any four days she chooses. • How many schedules can Janine choose from? • 7 • 6 • 5 • 4 1 • 2 • 3 • 4 • Combination: working M,T,W,TH is the same as working T,M,W,TH.

  35. Last one • Most days, you will teach Language Arts, Math, Social Studies, and Science. If Language Arts has to come first, how many different schedules can you make? • 1 • 3 • 2 • 1 • Permutation: the order of the schedule matters.

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