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ALGEBRA 2: 10.1 Apply the Counting Principle and Permutations Goal Use the fundamental counting principle and find permutations. VOCABULARY Permutation A permutation is an ordering of n objects. Factorial Represented by the symbol !, n factorial is defined as: n! = n {n 1) {n 2) .....3 2 1. FUNDAMENTAL COUNTING PRINCIPLE Two Events If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is __m n__ . Three or More Events The fundamental counting principle can be extended to three or more events. For example, if three events occur in m, n, and p ways, then the number of ways that all three events can occur is __m n p__. Example 1 Use the fundamental counting principle Pizza You are buying a pizza. You have a choice of 3 crusts, 4 cheeses, 5 meat toppings, and 8 vegetable toppings. How many different pizzas with one crust, one cheese, one meat, and one vegetable can you choose? Checkpoint Complete the following exercise. 1. If the pizza crust was not a choice in Example 1, how many different pizzas could be made? Example 2 Use the counting principle with repetition Telephone Numbers A town has telephone numbers that all begin with 329 followed by four digits. How many different phone numbers are possible (a) if numbers can be repeated and (b) if numbers cannot be repeated? Example 3 The standard configuration for a Texas license plate is 1 letter followed by 2 digits followed by 3 letters. a. How many different license plates are possible if letters and digits can be repeated? b. How many different license plates are possible if letters and digits cannot be repeated? Example 4 Find the number of permutations Playoffs Eight teams are competing in a baseball playoff. a. In how many different ways can the baseball teams finish the competition? b. In how many different ways can 3 of the baseball teams finish first, second, and third? PERMUTATIONS OF n OBJECTS TAKEN r AT A TIME The number of permutations of r objects taken from a group of n distinct objects is denoted by nPr n! P n r ( n r )! Example 5 Find permutations of n objects taken rat a time Homework You have 6 homework assignments to complete over the weekend. However, you only have time to complete 4 of them on Saturday. In how many orders can you complete 4 of the assignments? Checkpoint Complete the following exercises. 2. How many different 7 digit telephone numbers are possible if all of the digits can be repeated? 3. In Example 3, how many different ways can the teams finish if there are 6 teams competing in the playoffs? 4. You were left a list of 9 chores to complete. In how many orders can you complete 5 of the chores? PERMUTATIONS WITH REPETITION The number of distinguishable permutations of n objects where one object is repeated s± times, another is repeated s2 times, and so on, is: n! s1 ! s 2 s k ! Example 6 Find permutations with repetition Find the number of distinguishable permutations of the letters in (a) EVEN and (b) CALIFORNIA. Checkpoint Find the number of distinguishable permutations of the letters in the word. 5. TOMORROW 6. YESTERDAY 10.2 Use Combinations and the Binomial Theorem Goal Use combinations and the binomial theorem. VOCABULARY Combination A selection of r objects from a group of n objects where the order is not important Pascal's triangle An arrangement of the values of nCr w a triangular pattern in which each row corresponds to a value of n Binomial theorem For any positive integer n, the binomial expansion of (a + b)n is (a+ b)n = nC0anb0 + nC1an1b1 + nC2an2b2 + + nCna0bn COMBINATIONS OF n OBJECTS TAKEN r AT A TIME The number of combinations of r objects taken from a group of n distinct objects is denoted by nCr nCr = n! (n r )! r! Example 1 Find combinations Books You are picking 7 books from a stack of 32. If the order of the books you choose is not important, how many different 7 book groups are possible? Example 2 A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit. a. If the order in which the cards are dealt is not important, how many different 5-card hands are possible? b. In how many 5-card hands are all 5 cards of the same color? Example 3 Decide to multiply or add combinations Movie Rentals The local movie rental store is having a special on new releases. The new releases consist of 12 comedies, 8 action, 7 drama, 5 suspense, and 9 family movies. a. You want exactly 2 comedies and 3 family movies. How many different movie combinations can you rent? b. You can afford at most 2 movies. How many movie combinations can you rent? Example 4 William Shakespeare wrote 38 plays that can be divided into three genres. Of the 38 plays, 18 are comedies, 10 are histories, and 10 are tragedies. a. How many different sets of exactly 2 comedies and 1 tragedy can you read? b. How many different sets of at most 3 plays can you read? Checkpoint Complete the following exercises. 1. Find 7C4. 2. Find 6C3. 3. Find 12C11. 4. From Example 3, find the number of possible movie combinations if you can choose 2 action movies and 2 dramas. Example 5 Solve a multi-step problem Reading A popular magazine has 11 articles. You want to read at least 2 of the articles. How many different combinations of articles can you read? Example 6 During the school year, the girls’ basketball team is scheduled to play 12 home games. You want to attend at least 3 of the games. How many different combinations of games can you attend? Checkpoint Complete the following exercise. 5. Your school football team has 10 scheduled games for the season. You want to attend at least 4 games. How many different combinations of games can you attend? PASCAL'S TRIANGLE The first and last numbers in each row are _1_. Every number other than _1_ is the sum of the closest two numbers in the row directly above it. Pascal's triangle: As combinations n = 0 (0th row) 0C _0_ n = 1 (1st row) 1C_0_ 1C_1_ n = 2 (2nd row) 2C_0_ 2C_1_ 2C_2_ n = 3 (3rd row) 3C_0_ 3C_1_ 3C_2_ 3C_3_ As numbers _1_ _1_ _1_ _1_ _2_ _1_ _1_ _3_ _3_ _1_ Example 7 Use Pascal's triangle Class Representatives Out of 5 finalists, your class must choose 3 class representatives. Use Pascal's triangle to find the number of combinations of 3 students that can be chosen as representatives. Checkpoint Complete the following exercise. 6. In Example 4, use Pascal's triangle to find the number of combinations of 3 students that can be chosen from 8 finalists. BINOMIAL THEOREM For any positive integer n, the binomial expansion of (a + b)n is: (a + b)n = nC0anb0 + nC1an 1b1 + +nCna0bn Notice that each term in the expansion of (a + b)n has the form nCran 1rbr where r is an integer from 0 to n. Example 8 Expand a power of a binomial sum Use the binomial theorem to write the binomial expansion. (x + 4)3 Example 9 Expand a power of a binomial difference Use the binomial theorem to write the binomial expansion. (2m n)4 Example 10 Find a coefficient in an expansion Find the coefficient of x5 in the expansion of (2x 7)9. Checkpoint Use the binomial theorem to write the binomial expansion. 7. (a + 2b)3 8. (6 s)4 9. Find the coefficient of x8 in the expansion of (3x 2)10. 10.3 Define and Use Probability Goal Find the likelihood that an event will occur. VOCABULARY Probability A number from 0 to 1 that indicates the likelihood an event will occur Theoretical probability When all outcomes are equally likely, the number of outcomes in event A divided by the total number of outcomes Odds Used to measure the chances in favor of an event occurring or the chances against an event occurring Experimental probability A calculation of the probability of an event based on performing an experiment Geometric probability Probability found by calculating a ratio of two lengths, areas, or volumes THEORETICAL PROBABILITY OF AN EVENT When all outcomes are equally likely, the theoretical probability that an event A will occur is: Number of outcomes in event A P(A)= Total number of outcomes The theoretical probability of an event is often simply called the probability of the event. Example 1 You roll a standard six-sided die. Find the probability of (a) rolling a 5 and (b) rolling an even number. Checkpoint Complete the following exercises. 1. Using a standard deck of 52 playing cards, find the probability of drawing a red queen. 2. Using a standard deck of 52 playing cards, find the probability of a) picking an 8 and b) picking a red king. Example 2 You have an equally likely chance of choosing any integer from 1 through 20. Find the probability of the given event. a. A perfect square is chosen. b. A factor of 30 is chosen. ODDS IN FAVOR OF OR ODDS AGAINST AN EVENT When all outcomes are equally likely, the odds in favor of an event A and the odds against an event A are: Number of outcomes in A Odds in favor of event A = Number of outcomes not in A Number of outcomes not in A Odds against event A = Number of outcomes in A a You can write the odds in favor of or against an event in b the for or in the form a:b Example 3 Find odds Marbles A marble is drawn from a bag containing 6 red, 12 yellow, and 9 black marbles. Find (a) the odds in favor of drawing a red marble and (b) the odds against drawing a black marble. EXPERIMENTAL PROBABILITY OF AN EVENT When an experiment is performed that consists of a certain number of trials, the experimental probability of an event A is given by: Number of trials where A occurs P(A) = Total number of trials Example 4 Find an experimental probability Exam Grades Exam grades of students in a history class are shown in the bar graph. Find the probability that a randomly chosen student in this history class received a C or better. 1. A marble is drawn from a bag containing 6 blue, 6 yellow, 9 white, and 11 green marbles. Find the odds in favor of drawing a yellow marble. 2. Using Example 4, find the experimental probability that a randomly selected student received a C or lower. 10.4 Find Probabilities of Disjoint and Overlapping Events Goal Find probabilities of compound events. VOCABULARY Compound Event The union or intersection of two events Disjoint or mutually exclusive events Two events that have no outcomes in common PROBABILITY OF COMPOUND EVENTS If A and B are two events, then the probability of A or B is: P(A or B) = _P(A) + P(B) P(A and B)_ If A and B are disjoint events, then the probability of A or B is: P(A or B) = _P(A) + P(B)_ Example 1 Find probability of disjoint events You roll a six-sided number cube. What is the probability of rolling a 2 or a 5? Checkpoint Complete the following exercise. 1. You roll a six-sided number cube. What is the probability of rolling a 1 or an even number? Example 2 Find probability of compound events You roll a six-sided number cube. What is the probability of rolling an odd number or a number less than 3? Example 3 Use a formula to find P(A and B) Music In a survey of 300 students, 150 like pop music or country music. There are 97 students who like pop music and there are 83 students who like country music. What is the probability that a randomly selected student likes both pop music and country music. Example 4 Find the probability of complements When two six-sided dice are rolled, there are 36 possible outcomes. Find the probability that the sum is not 4 and the sum is greater than or equal to 3. PROBABILITY OF THE COMPLEMENT OF AN EVENT The probability of the complement of A is P( A ) = _1 P(A)_. Checkpoint Complete the following exercises. 2. You roll a six-sided number cube. What is the probability of rolling a number less than 4 or an even number? 3. In a survey of 125 people, 90 of them like orange juice or grape juice. There are 62 people who like orange juice and 43 people who like grape juice. What is the probability that a randomly selected person likes both orange juice and grape juice? 4. From Example 4, find the probability that the sum is not 8. 10.5 Find Probabilities of Independent and Dependent Events Goals Examine independent and dependent events. VOCABULARY Independent events Two events such that the occurrence of one has no effect on the occurrence of the other Dependent events Two events such that the occurrence of one affects the occurrence of the other Conditional probability The probability that event B will occur given that event A has occurred is called the conditional probability of B given A. PROBABILITY OF INDEPENDENT EVENTS If A and B are independent events, then the probability that both A and B occur is: P(A and B) = _P(A) P(B)_ More generally, the probability that n independent events occur is the _product_ of the n probabilities of the individual events Example 1 Find probability of three independent events Attendance Every morning, one student in a class of 24 students is randomly chosen to take attendance. What is the probability that the same student will be chosen three days in a row? Checkpoint Complete the following exercises. 1. During a high school track meet, each race consists of 9 competitors who are randomly assigned lanes from 1 to 9. What is the probability that a runner will draw lanes 1, 2, or 3 in the three races in which he competes? PROBABILITY OF DEPENDENT EVENTS If A and B are dependent events, then the probability that both A and B occur is: P(A and B) = __P(A) P(B A)__ Example 3 Find a conditional probability Eye Color Find the probability that (a) a listed person has blue eyes and (b) a male has blue eyes. Green eyes Blue eyes Brown eyes Hazel eyes Male 27 35 15 23 Female 12 9 38 41 Example 4 Comparing independent and dependent events You randomly select two marbles from a bag containing 15 yellow, 10 red, and 12 blue marbles. What is the probability that the first marble is yellow and the second marble is not yellow if (a) you replace the first marble before selecting the second, and (b) you do not replace the first marble? Example 5 Find probability of three dependent events Pencils Your teacher passes around a box with 10 red pencils, 8 pink pencils, and 13 green pencils. If you and the two people in your group are the first to randomly select a pencil, what is the probability that all three of you select pink pencils? Checkpoint Complete the following exercises. 2. Use the table in Example 3 to find the probability that a female has hazel eyes. 3. From Example 4, find the probability that both marbles will be yellow if (a) you replace the first marble and (b) you do not replace the first marble. 4. From Example 5, what is the probability that you and your group members all choose a red pencil? 11.1 Find Measures of Central Tendency and Dispersion Goal Describe data using statistical measures. VOCABULARY Statistics Numerical values used to summarize and compare sets of data Measure of central tendency A number used to represent the center or middle of a set of data values. This is represented by the mean, median, and mode. Measure of dispersion A statistic that tells you how dispersed, or spread out, data values are Standard deviation A measure that describes the typical difference (or deviation) between a data value and the mean Outlier A value that is much greater than or much less than most of the other values in a data set MEASURES OF CENTRAL TENDENCY The mean, or __average__ , of n numbers is the __sum__ of the numbers __divided__ by n. The mean is denoted by x , which is read as "x-bar." For the data set x1,x2,…xn, the mean is x x 1 x 2 ... xn n . The median of n numbers is the __middle__ number when the numbers are written in order. (If n is even, the median is the __mean__ of the two middle numbers.) The mode of n numbers is the number or numbers that occur __most frequently__. There may be _one_ mode, __no__ mode, or __more than one__ mode. Example 1 Find measures of central tendency Quiz Scores The data sets at the right give quiz scores for two different biology classes. Find the mean, median, and mode of each data set. Class A Class B 15, 17, 17, 17, 18, 19, 21, 22, 25 16, 18, 19, 21, 22, 22, 22, 24, 25 STANDARD DEVIATION OF A DATA SET The standard deviation (read as "sigma") of x1, x2,…xn is: = x 1 2 2 x x 2 x ... xn x 2 n Example 2 Find the range and standard deviation Find the range and standard deviation for the quiz scores in each data set from Example1. Example 3 Examine the effect of an outlier Soccer The winning scores for the first 9 games of the soccer season are: 3, 4, 2, 5, 3,1, 4, 3, 2. a. Find the mean, median, mode, range, and standard deviation of the data set. b. The winning score in the next game is an outlier, 9. Find the new mean, median, mode, range, and standard deviation. c. Which measure of central tendency does the outlier affect the most? the least? d. What effect does the outlier have on the range and standard deviation? Checkpoint Complete the following exercises. The data set below gives the recorded speeds (in mi/h) for 10 different cars on a local highway. 69, 62, 64, 67, 62, 64, 63, 65, 60, 64 1. Find the range and standard deviation of the data set. 2. Find the range and standard deviation of the data set 3. The next car that drives by is having car trouble, so the recorded speed is 36 mi/h. Find the new mean, median, mode, range, and standard deviation.