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12-1: The Counting Principle Learning Targets:

12-1: The Counting Principle Learning Targets:. I can distinguish between independent and dependent events. I can solve problems involving independent and dependent events. The Counting Principle Definitions. trial : an experiment (like flipping a coin)

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12-1: The Counting Principle Learning Targets:

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  1. 12-1: The Counting PrincipleLearning Targets: • I can distinguish between independent and dependent events. • I can solve problems involving independent and dependent events.

  2. The Counting PrincipleDefinitions trial: an experiment (like flipping a coin) outcome: the result of a single trial sample space: a list of all possible outcomes event: one or more outcomes of a trial

  3. The Fundamental Counting Principle If event M can occur in m ways and event N can occur in n ways, then event M followed by event N can occur in m•n ways. • works with dependent events • works with independent events

  4. Dependent and IndependentEvents independent: the outcome of one event does not impact the outcome of another event (rolling a die or tossing a coin) dependent: the outcome of one event does impact the outcome of another event (taking a sock out of a drawer and then taking another sock out of the same drawer without replacement of the first one)

  5. Independent Events A sandwich menu offers customers a choice of white, rye, or cheese bread with one spread chosen from butter, mustard, or mayonnaise. How many differentcombinations of bread and spread are there? Make a tree diagram to see your sample space: White Rye Cheese Bread Butter Mustard Mayo Butter Mustard Mayo Butter Mustard Mayo Nine possible combinations. You could also do: Number of Breads ● Number of Spreads = 3 ● 3 = 9

  6. Independent Events A pizza place offers customers a choice of American,mozzarella, Swiss, feta, or provolone cheese with onetopping chosen from pepperoni, mushrooms orsausage. How many different combinations of cheeseand toppings are there? Make a tree diagram to see your sample space: Give a few minutes to complete.

  7. Independent Events – Tree Diagram American Mozzarella Swiss Feta Provolone Pep. Mush. Sau. Pep. Mush. Sau. Pep. Mush. Sau. Pep. Mush. Sau. Pep. Mush. Sau. 15 possible combinations. You could also do: Number of Cheeses ● Number of Toppings = 5 ● 3 = 15

  8. More Than Two Independent Events CommunicationHow many codes are possible if ananswering machine requires a 2-digit code to retrieve messages? Two Digit Code: ____ ____ How many digits can you choose from for each spot? 10 10

  9. More Than Two Independent Events CommunicationHow many codes are possible if ananswering machine requires a 2-digit code to retrieve messages? Two Digit Code: _10_●_10_ There are 100 different codes to choose from.

  10. Possibilities Digits 0-9 = 10 Letters A-Z = 26 Cards 52 total 4 suits - 13 cards per suit

  11. Dependent Events How many different schedules could a student havewho is planning to take 4 different classes? Assumeeach class is offered each period. First Period Choices ● Second Period Choices● Third Period Choices● Fourth Period Choices If a class is choosen for first hour, it can not been choosen again.

  12. Dependent Events How many different schedules could a student havewho is planning to take 4 different classes? Assumeeach class is offered each period. 4 ● 3 ● 2 ● 1 = 24

  13. Assignment Work on the 21 problems that follow in the note packet.

  14. Algebra 2A - Chapter 12Section 2 Permutations and Combinations

  15. 12-2: Permutations and CombinationsLearning Targets: • I can solve problems with permutations. • I can solve problems with combinations.

  16. Permutations permutation: when a group of objects or people are arranged in a certain order • order of objects very important The number of permutations of n distinct objects taken r at a time is given by

  17. Permutations Eight people enter the Best Pic contest. How many ways can blue, red, and green ribbons be awarded? Order Matters!!!

  18. Permutations How many permutations of the letters MATH are possible? Order Matters!!!

  19. Permutations How many different four-letter code words can be formed from the word EQUATIONS ? Order Matters!!! Also known as factorial:

  20. Permutations with Repetition The number of permutations of n objects of which p are alike and q are alike is: How many different ways can the letters of the wordBANANA be arranged? You will notice some repetition here. The letter A appears thrice and the letter N appears twice.

  21. Combinations combination: an arrangement or selection of objects in which order is not important The number of combinations of n distinct objects taken r at a time is given by

  22. Combinations Five cousins at a family reunion decide that three ofthem will go to pick up a pizza. How many ways canthey choose three people to go? Order Does Not Matters!!!

  23. Combinations There are 60 players on a football team. Seven of them will be chosen for a random drug test. How many ways can they be chosen? Order Does Not Matters!!!

  24. Multiple Events Six cards are drawn from a standard deck of cards.How many hands consist of two hearts and four spades? Order Does Not Matters!!! There are 13 cards per suit. Hearts Spades

  25. Multiple Events Thirteen cards are drawn from a standard deck of cards. How many hands consist of six hearts and seven diamonds? Order Does Not Matters!!! There are 13 cards per suit. Hearts Diamonds

  26. How many of you have parents that play the Lottery? Let’s calculate the number of different combinations there possibly are. Mega Millions Total Combinations   Since the total number of combinations for Mega Millions numbers is used in all the calculations, we will calculate it first. The number of ways 5 numbers can be randomly selected from a field of 56 is: COMBIN(56,5) = 3,819,816.    For each of these 3,819,816 combinations there are COMBIN(46,1) = 46 different ways to pick the sixth number (the “Mega” number). The total number of ways to pick the 6 numbers is the product of these. Thus, the total number of equally likely Mega Millions combinations is 3,819,816 x 46 = 175,711,536.

  27. for Understanding How many different ways can the letters of the wordALGEBRA be arranged? Six friends at a party decide that three of them will goto pick up a movie. How many ways can they choose three people to go? Ten people are competing in a swim race where 4ribbons will be given. How many ways can blue,red, green, and yellow ribbons be awarded?

  28. Assignment p. 641: 4-32

  29. Reflect A class of 250 students wants to elect a committee of 4 to buy supplies for the homecoming float. How many different committees are possible?

  30. Algebra 2A - Chapter 12Section 3 Probability

  31. 12-3: ProbabilityLearning Targets: • I can find probability and odds of events. • I can create and use graphs of probability distributions.

  32. Probability success: desired outcome failure: any outcome that is not a success If an event can succeed in s ways and fail in f ways, then the probabilities of success P(S), and of failure, P(F), are as follows:

  33. Probability If an event can succeed in s ways and fail in f ways, then the probabilities of success P(S), and of failure, P(F), are as follows: Probability is between 0 and 1, inclusive. The closer to 1, the more likely the event is to occur. The closer to 0, the less likely the event is to occur.

  34. Probability When two coins are tossed, what is the probability that both are tails? Use a sample space, tree diagram: Toss #1: H T Toss #2: H T H T

  35. Probability with Combinations Monica has a collection of 32 CDs, of which 18 are R&B and 14 are rap. As she’s leaving for a trip, she grabs 6 CDs. What is the probability that she selects 3 R&B and 3 rap? Combinations of Rap Combinations of R&B

  36. Probability with Combinations Roman has a collection of 26 books–16 are fiction and10 are nonfiction. He randomly chooses 8 books to take with him on vacation. What is the probability thathe chooses 4 fiction and 4 nonfiction?

  37. Odds The odds that an event will occur can be expressed as the ratio of the number of ways it can succeed to the number of ways it can fail. If an event can succeed in s ways and fail in f ways, then the odds of success and of failure are as follows: Odds of success = s : f Odds of failure = f : s Notice: s + f = Total Possibilities

  38. Odds According to the CDC, the chances of a male born in 1990 living to age 65 are about 3 in 4. For females the chances are about 17 in 20. What are the odds of a male living to be at least 65? What are the odds of a female living to be at least 65? 3:1 Success Failure: 4 -3 17:3

  39. Probability DistributionsWhich outcomes are least likely? most likely? 2 S = Sum 3 4 5 6 7 8 9 10 11 12 Probability Suppose two dice are rolled. The table and the relative-frequency histogram show the distribution ofthe sum of the numbers rolled.

  40. for Understanding When three coins are tossed, what is the probabilitythat all three are heads? Life ExpectancyThe chances of a male born in 1980to live to be at least 65 years of age are about 7 in 10.For females, the chances are about 21 in 25. Calculate the odds for each sex living at least 65 years.

  41. Assignment p. 647: 4-18 p. 648: 19-53

  42. Reflect If 7 out of 8 students prefer the subject of math to literature, what are the odds that students prefer math? that students prefer literature?

  43. Algebra 2A - Chapter 12Section 4 Multiplying Probabilities

  44. 12-4: Multiplying ProbabilitiesLearning Targets: • I can find the probability of two independent events. • I can find the probability of two dependent events.

  45. Probability Rules Probability of two independent events: P(A and B) = P(A) • P(B) Probability of two dependent events: P(A and B) = P(A) • P(B following A) extends to P(A, B, C) = P(A) • P(B following A) • P(C following A and B)

  46. Independent Events At a picnic Julio reaches into an ice-filled cooler containing 8 regular and 5 diet soft drinks. He removes a can, then decides he is not really thirsty, so he puts it back. What is the probability that Julio and the next person to reach into the cooler both randomly select a regular soft drink? This is a problem With Replacement!!

  47. Independent Events Gernardo has 9 dimes and 7 pennies in his pocket. He randomly selects one coin, looks at it, and replaces it. He then randomly selects another coin. What is the probability that both of the coins he selects are dimes? This is a problem With Replacement!!

  48. Independent Events Extended In a board game, three dice are rolled to determine the number of moves for the players. what is the probability that the first die shows a 6, the second die shows a 6, and the third die does not? P(6) P(6) P(not 6) =

  49. Three Independent Events When three dice are rolled, what is the probabilitythat two dice show a 5 and the third die shows aneven number? P(5) P(5) P(even) =

  50. Two DEPENDENT Events In the previous Julio and the soft drink example, what is the probability that both people select a regular soft drink if Julio does NOT put his drink back into the cooler? This is a problem Without Replacement!!

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