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 Combinations vs. Permutations

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…?

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

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.

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.

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

Here’s a harder one… Suppose I spin the spinner. B R Y W Suppose 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 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.

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?

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?

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

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?

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 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.

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

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

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

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.

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.

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.

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.

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?

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.

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.

Divide by 9! (to get rid of duplicates). Write it this way: 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: 11,440 Permutations: 4,151,347,200 Since the batting order does matter, this is an example of a permutation.

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)

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

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?

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)

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 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.

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.

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.