suppose i have two fair dice. player one gets 2 points if the sum is odd. player two gets 4 points...

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  • 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?

  • AgendaReview finding probability Determine expected valueIs this game fair--1 player? 2 players?Fundamental Counting PrincipleCombinations vs. Permutations

  • Expected valueExpected 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?

  • Expected valueIn general, I consider each event that is possible in my experiment. Each event has its 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

  • Here are three easy examplesRoll a 6-sided die. If you roll a 3, then you win $5.00. If you dont roll a 3, then you have to pay $1.00.P(3) = 1/6P(not 3) = 5/6P(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.

  • Here are three easy examplesRoll 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) = 1/2Expected value(1/3)(.25) + (1/6)(1) +(1/2)(-.50) = .0833 + .1667 -.25 = 0. Another fair game.

  • Here are three easy examplesIs 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

  • Heres a harder oneSuppose I spin the spinner.Here are the rules. If I spin blue or white, I get a quarter. If I spin red, I get a nickel. If I spin yellow, 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

  • One eventOn 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.

  • Two eventsI 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?

  • Two eventsThere 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?

  • Watch the wordingSuppose I flip a coin.P(H) =P(T) =P(H or T) =P(H and T) =

  • True/FalseSuppose you have a true/false section on tomorrows 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?

  • 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

  • Fundamental Counting PrincipleSo, 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.

  • Fundamental Counting PrincipleSuppose 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

  • Fundamental Counting PrincipleNow, 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

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

  • For dependent events, Permutations vs. CombinationsIn 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.

  • ExamplesI 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.

  • Permutations and CombinationsIn 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.

  • Combinations: dont count duplicatesSo, how do I get rid of the duplicates?Lets 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.

  • 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 DBuild from ABC: DABC, ADBC, ABDC, ABCDNow build from ACB:DACB, ADCB, ACDB, ACBDKeep going How many possible?

  • FactorialSo, 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.

  • Two more practice problemsSuppose 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 (dont consider the position). Solve.Permutation or combination: kids batting order. Solve.

  • 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

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

  • Another exampleMy 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)

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

  • Deal or no DealYou 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?

  • A few practice problemsA 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)

  • 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.

  • Review Permutations and CombinationsI 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 4This is a permutation.

  • Review permutations and combinationsJanines 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 4Combination: working M,T,W,TH is the same as working T,M,W,TH.

  • Last oneMost 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 1Permutation: the order of the schedule matters.

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