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Chapter 12 Notes Algebra 2 Name ____________________________________ Period _____ 12.1 The Fundamental Counting Principle and Permutations Ex. 1 How many ways can you make a salad given: 2 types of lettuce: romaine and iceberg and four types of dressing: ranch, 1000, blue cheese, and Italian. Tree Diagram: Fundamental Counting Principle: 2 or more events = o 2: if an event occurs m ways and another event occurs n ways, then the number of ways that both can occur is m n . o 3 or more: the product of all the number of ways is m n o ... z . Ex. 2 At a restaurant, you have a choice of 8 different entrees, 2 different salads, 12 different drinks, and 6 different desserts. How many different dinners, given one of each, can you choose? Ex. 3 Consider a standard license plate with 7 items. If the first 4 items were numbers and the last 3 items were letters: A) How many different plates are possible if both numbers and letters can repeat? B) How many different plates are possible if both numbers and letters can’t repeat? Ex. 4 How many different 7 digit phone numbers are possible if the first digit cannot be 0 or 1? Permutation: an ordering of n objects n! n n 1 n 2 n 3 ... 3 2 1 o ie: number of ways to arrange the numbers 1, 2, 3 = 3! 3 2 1 6 ways Definition: n ! 1 2 3 4 ... n and 0! 1 . Ex. 5 You have homework assignments from 5 different classes to complete this weekend. A) In how many different ways can you complete the assignments? B) In how many different ways can you choose 2 of the assignments to complete first and last? For part B: the number of permutations of 5 objects taken 2, is 5! denoted by 5 P2 and given by . 5 2 ! * Permutations of n objects taken r at a time: (________________ matters!) n Pr ! ! Ex. 6 There are 12 books on the summer reading list. You want to read some or all of them. In how many orders can you read A) 4 of the books? B) All 12 of the books? During a permutation, the objects are not always distinct. Some objects may repeat. o Ie: mom: mom omm mmo m o m, o m m, m m o mom omm mmo * Permutations with Repetition: where one object is repeated q1 times, another is repeated q2 times and so on. n! q1 ! q2 ! ... qk ! Ex. 7 Find the number of distinguishable permutations of the letters in A) Summer B) Waterfall C) Mississippi D) Word 12.2 Combinations and the Binomial Theorem Combination: is a selection of r objects from a group of n objects where order is _________ important. * Combination: (order does _______ matter) ie. a,b,c and b,c,a are the same n Cr ! ! ! Ex. 1 Use a standard deck of 52 cards. A) If order is not important, how many different 7-card hands are possible? B) How many of these 7-card hands have all 7 cards of the same suit? C) How many 5 card hands contain exactly 3 kings? Number of ways: o event a ___________ event b occurs You _______________________ o event a ________ even b occurs You ___________ Ex. 2 You are taking a vacation. You can visit as many as 5 different cities and 7 different attractions. A) Suppose you want to visit exactly 3 different cities and 4 different attractions. How many different trips are possible? B) Suppose you want to visit at least 8 locations (cities or attractions). How many different types of trips are possible? Ex. 3 A restaurant offers 6 salad toppings. On a deluxe salad, you can have up to 4 toppings. How many different combinations of toppings can you have? 4 Pascal’s Triangle: an arrangement of n C r : 0 C0 1 C0 1 C1 2 C0 2 C1 2 3 C0 3 C1 C0 4 C1 4 3 C2 C2 C2 4 3 C3 C3 4 C4 12.2 (Part 2) Binomial Theorem: the expansion of a b : n a b n C 0 an b0 n C 1 an 1 b1 n C 2 an 2 b2 ... n C n a0 bn n n n C r a n r br r 0 Ex. 4 Expand a 3 5 Ex. 5 Expand a 2b3 4 Ex. 6 Expand x 5 4 Ex. 7 Expand 2x y 2 3 Ex. 8 Find the coefficient of x7 in 2 3x . 10 Ex. 9 Find the coefficient of x4 in 2 x 3 . 12 12.3 An Introduction to Probability Probability: a number between ___ and ___ that indicates the likelihood the event will occur (answer can be written is fraction, decimal, or percent). o probability = what you ___________ over the _____________ thing. P = ___: event can’t occur P = ___: event always occurs o Theoretical Probability: when all outcomes are _______________ likely. o P a number of outcomes in A total number of outcomes o Experimental Probability: by an experiment, survey, or looking at the history of the event. o Geometric Probability: calculating a ratio of 2 lengths, areas, or volumes. Ex. 1 A spinner has 8 equal-size sectors numbered from 1 to 8. Find the probability of: A) spinning a 6 B) spinning a number greater than 6. Ex. 2 There are 9 students on the math team. You draw their names one by one to determine the order in which they answer questions at a math meet. What is the probability that 3 of the 5 seniors on the team will be chosen last, in any order? Ex. 3 Find the probability that a randomly chosen student from this year’s 9th grade class is enrolled in: A) Consumer Math B) Algebra 1 or Introduction to Algebra Ex. 4 Find the probability that a dart would hit the shaded region. Ex. 5 Five cards are drawn from a 52-card deck. What is the probability the first two cards are red? Ex. 6 Find the probability of drawing the given card from a standard 52-card deck: a) the queen of hearts b) a non-face card 12.4 Probability of Compound Events Compound Event: the union or intersection of 2 events. o Union: all outcomes of ____________ A and B. o Intersection: only _______________ outcomes of A and B. Mutually Exclusive Events: when there is ______ outcomes in the intersection of A and B. Probability of Compound Events: o or and P A B P A P B P A B o P A B P A P B no intersection (mutually exclusive) P A B 0 Ex. 1 One six-sided die is rolled. What is the probability of rolling a multiple of 2 or a 5. Ex. 2 A card is randomly selected from a standard deck of 52 cards. What is the probability that the card is a heart or a number less than 3 (including the ace)? Ex. 3 In a poll of high school juniors, 6 out of 15 took a French class and 11 out of 15 took a math class. Fourteen out of 15 students took French or math. What is the probability that a student took both French and math? Complement of Event A: (A’) are all outcomes that are ______ in A. (A prime). Probability of the Complement of an Event: o P( A ') ______________ Ex. 4 A card is randomly selected from a standard deck of 52 cards. Find the probability of the given event. A) The card is not a king B) The card is not an ace or a jack Ex. 5 When 2 six-sided dice are tossed, there are 36 possible outcomes. Find the probability of the given event (look on page 726 for the possible outcomes). A) The sum is not 5 B) The sum is greater than or equal to 3 12.5 Probability of Independent and Dependent Events Independent: when the first event has _________________ to do with the second event. (ie: rolling a die twice). Probability of Independent Events: o and P A B P A P B Dependent: one event _______________ the occurrence of the other. Conditional Probability: the probability that b will occur _____________ that A has occurred. “B given A” P B A Probability of Dependent Events: o P A B P A P B A Where P B A P A B P A Ex. 1 A game machine claims that 1 in every 15 people win. What is the probability that you win twice in a row? Ex. 2 In a survey 9 out of 11 men and 4 out of 7 women said they were satisfied with a product. A) If the next 3 customers are 2 women and a man, what is the probability that they will all be satisfied? B) If 4 men are the next customers, what is the probability that at least one of them is not satisfied? Ex. 3 You randomly select 2 cards from a standard 52-card deck. Find the probability that the first card is a diamond and the second card is red if A) You replace the 1st card before selecting the 2nd card. B) You don’t replace the first card. Ex. 4 Age 5–7 8 – 10 11 – 13 Total Attended Camp 45 94 81 No Camp 117 62 79 Total A) Find the probability that a listed student attended camp B) Find the probablity that a child in the 8 – 10 age group from the town did not attend camp. Ex. 5 Three children have a choice of 12 summer camps that they can attend. If they randomly choose which camp to attend, what is the probability that they attend all different camps? Ex. 6 In one town 95 % of the students graduate from high school. Suppose a study showed that at age 25, 81 % of the high school graduates held full-time jobs, while only 63 % of those who did not graduate held full-time jobs. What is the probability that a randomly selected student will have a full-time job at 25? Ex. 7 A survey showed that 47 % of students worked during the summer. Of those who worked, 62 % watched 2 hours or more of TV per day. Those who didn’t work, 79 % watched 2 or more hours of TV. What is the probability that you choose a random student who watched fewer than 2 hours of TV per day. Ex. 8 One in 10,000 cars has a defect. How many of these cars can a car dealer sell before the probability of selling at least one with a defect is 20%? 12.6 Binomial Distributions Binomial Experiment: o There are a _______________ number, n, of independent trials. o Each trial has only ___ possible outcomes…success and failure o The probability of success is the ________ for each trial. This probability is denoted by p. The probability of failure is given by 1 – p. Finding a Binomial Probability: o P k successes n C k p k (1 p)n k Ex. 1 A) At a college, 53 % of students receive financial aid. In a random group of 5 students, what is the probablity that exactly 2 of them receive financial aid? B) Draw a histogram of the binomial distribution for the class of students. Then find the probability that fewer than 3 students in the class receive financial aid. Symmetric: when the histograms left side __________________ the right side. Skewed: ________ symmetric. Ex. 2 Kobe Bryant’s free throw percentage is 0.856. What is the probability he will miss 4 free throws in a game where he shoots 12 free throws. Hypothesis Testing: o State the hypothesis you are testing. o Collect data from a random sample of the population and compute the statistical measure of the sample. o Assume the hypothesis is true and calculate the resulting probability of obtaining the sample statistical measure. If the probability is small ( 10 % hypothesis. or 0.1 ) – reject the Ex. 3 A diet company claims that 78 % of people who use their product lose weight. To test this hypothesis, you randomly choose 9 people and ask them to do the diet for 3 months. Of the 9 people, only 4 lost weight. Should you reject this claim?