Download Chapter 9 Discrete Mathematics 9.1 Basic Combinatorics The

Chapter 9 Discrete Mathematics 9.1 Basic Combinatorics The Importance of Counting: In how many different ways can three distinguishable objects be arranged in order? The Multiplication Principle of Counting: If a procedure P has a sequence of stages S1, S2, …. Sn and if S1 can occur in r1 ways, S2 can occur in r2 ways, … Sn can occur in rn ways, Then the number of ways that the procedure P can occur is the product r1r2r3….rn The Tennessee license plate shown here consists of three letters of the alphabet followed by three numerical digits (0 through 9). Find the number of different license plates that can be formed a) if there is no restriction on the letters or digits that can be used b) if no letter or digit can be repeated One important application of the Multiplication Principle of Counting is to count the number of ways that a set of n objects (called an n-­‐set) can be arranged in order. Each such ordering is called a permuatation. Permutations of an n-­‐set: There are n! permutations of an n-­‐set. Count the number of different 9-­‐letter “words”(Don’t worry about whether they’re in the dictionary) that can be formed using the letters in each word: DRAGONFLY and BUTTERFLY. Distinguishable Permutations: There are n! distinguishable permutations of and n-­‐set containing n distinguishable objects. If an n-­‐set contains n1 objects of a first kind, n2 objects of a second kind, and so on with n1 + n2 + … + nk = n, then the number of distinguishable permutations of the n-­‐set is 𝑛!
𝑛! ! 𝑛! ! 𝑛! ! ⋯ 𝑛! !
Permutation Counting Formula: the number of permutations of n objects taken r at a time is denoted nPr and is given by !!
for 0 < r < n !!! !
If r > n, then nPr = 0. Evaluate without a calculator: 6P4 and nP3 Sixteen actors answer a casting call to try out for roles as dwarfs in a production of Snow White and the Seven Dwarfs. In how many different ways can the director cast the seven roles? Combination Counting Formula: The number of combinations of n objects taken r at a time denoted nCr and is given by !!
for 0 < r < n !! !!! !
If r > n then nCr = 0. In each of the following scenarios, tell whether permutations (ordered) or combinations (unordered) are being described. a) A president, vice-­‐president, and secretary are chosen from a 25-­‐member club. b) A cook chooses 5 potatoes from a bag of 12 potatoes to make a potato salad. c) A teacher makes a seating chart for 22 students in a classroom with 30 desks. In the Miss America pageant, 51 contestants must be narrowed down to 10 finalists who will compete on national television. In how many possible ways can the ten finalists be selected? The Georgia Lotto requires winners to pick 6 integers between 1 and 46. The order in which you select them does not matter; the lottery tickets are always printed with the numbers in ascending order. How many different lottery tickets are possible? Formula for Counting Subsets of an n-­‐Set: There are 2n subsets of a set with n objects (including the empty set and the entire set). A national hamburger chain used to advertise that it fixed its hamburgers “256 ways” since patrons could order whatever toppings they wanted. How many toppings must be available? Assignment page 708 # 3-­‐24 (3s) 9.2 The Binomial Theory Binomial Coefficient: The binomial coefficients that appear in the expansion of (a+b)n are the values of nCr for r = 0, 1, 2, 3, …, n. 𝑛
A classical notation for nCr especially in the context of binomial coefficients, is ( ). 𝑟
Both notations are read “n choose r”. Pascal’s Triangle Show how row 5 of Pascal’s triangle can be used to obtain row 6, and use the information to write the expansion of (x + y)6. 𝑛
𝑛−1
𝑛−1
Recursion Formula for Pascal’s Triangle: =
+
or nCr = n-­‐1Cr-­‐1 + n-­‐
𝑟
𝑟−1
𝑟
1Cr. Find the coefficient of x10 in the expansion of (x+2)15. Binomial Theorem: For any positive integer n, 𝑛 !
𝑛 !!!
𝑛
𝑛
(𝑎 + 𝑏)! =
𝑎 +
𝑎 𝑏 + ⋯ + ( )𝑎!!! 𝑏 ! + ⋯ + ( )𝑏 ! 0
1
𝑟
𝑛
where 𝑛
!!
( ) = nCr = !! !!! ! 𝑟
Expand (2x -­‐ y2)4. Basic Factorial Identities: For any integer n > 1, n! = n(n-­‐1)! For any integer n > 0, (n+1)! = (n+1)n! 𝑛
𝑛+1
Prove that (
) – ( ) = n for all integers n > 2. 2
2
Assignment page 715 #3-­‐24 (3s) page 708 # 27 9.3 Probability Find the probability of each of the following events: a) tossing a head on one toss of a fair coin b) tossing two heads in a row on two tosses of a fair coin c) drawing a queen from a standard deck of 52 cards d) rolling a sum of four on a single roll of two fair dice e) guessing all 6 numbers in a state lottery that requires you to pick 6 numbers between 1 and 46 inclusive. Probability of an Event (Equally Likely Outcomes) If E is an event in a finite, nonempty sample space S of equally likely outcomes, then the probability of the event E is !!! !"#$%& !" !"#$!%&' !" !
P(E) = !!! !"#$%& !" !"#$!%&' !" ! Find the probability of rolling a sum divisible by 3 on a single roll of two fair dice. A probability function is a function P that assigns a real number to each outcome in a sample space S subject to the following conditions: 1. 0 < P(O) < 1 for every outcome, O 2. the sum of the probabilities of all outcomes in S is 1 3. P(∅) = 0 Probability of an Event (Outcomes not Equally Likely): Let S be a finite, nonempty sample space in which every outcome has a probability assigned to it by a probability function P. If E is any event in S, the probability of the event E is the sum of the probabilities of all the outcomes contained in E. Is it possible to weight a standard 6-­‐sided die in such a way that the probability of rolling each number n is exactly 1/(n2+1)? Sal opens a box of a dozen chocolate crèmes and generously offers two of them to Val. Val likes vanilla crèmes the best, but all the chocolates look alike on the outside. If four of the twelve crèmes are vanilla, what is the probability that both of Val’s picks turn out to be vanilla? Multiplication Principle of Probability: Suppose an event A has probability p1 and an event B has probability p2 under the assumption that A occurs. Then the probability that both A and B occur is p1p2. Sal opens a box of a dozen chocolate crèmes and generously offers two of them to Val. Val likes vanilla crèmes the best, but all the chocolates look alike on the outside. If four of the twelve crèmes are vanilla, what is the probability that both of Val’s picks turn out to be vanilla. Geometric models can help us understand algebraic models more easily. Venn diagrams, associated mainly with set theory, are good for visualizing relationships among events within sample spaces. Tree diagrams, a way to visualize the Multiplication Principle of Counting, are good for visualizing the Multiplication Principle of Probability. In a large high school, 54% of the students are girls and 62% of the students play sports. Half of the girls at the school play sports. a) What percentage of the students who play sports are boys? b) If a student is chosen at random, what is the probability that it is a boy who does not play sports? Two identical cookie jars are on a counter. Jar A contains 2 chocolate chip and 2 peanut butter cookies, while jar B contains 1 chocolate chip cookie. We select a cookie at random. What is the probability that it is a chocolate chip cookie? Conditional Probability: If the event B depends on the event A, then P(B|A) = !(! & !)
!(!)
. Suppose we have to drawn a cookie at random from one of the jars described in our last example. Given that it is chocolate chip, what it the probability that it came fro jar A? We roll a fair die four times. Find the probability that we roll: a) all 3’s. b) no 3’s c) exactly 2 3’s. Binomial Distribution: Suppose an experiment consists of n independent repetitions of an experiment with two outcomes called “success” and “failure”. Let P(success) = p and P(failure) = q (Note that q = 1 – p). Then the terms in the binomial expansion of (p+q)n give the respective probabilities of exactly n, n-­‐1, …, 2, 1, 0 successes. Number of successes out of n Probability Independent repetitions. n pn 𝑛
n-­‐1 (
)pn-­‐1q 𝑛−1
… … 𝑛 r n-­‐r
r ( )p q 𝑟
… … 𝑛
1 ( )pqn-­‐1 1
0 qn Suppose Michael makes 90% of his free throws. If he shoots 20 free throws, and his chance of making each one is independent of the other shots. What is the probability that he makes a) all 20? b) Exactly 18? c) At least 18? Assignment page 728 # 3-­‐24(3s) page 715 #27 page 708 # 30 9.4 Sequences Find the first 6 terms and the 100th term of the sequence {ak} in which ak = k2-­‐1. Find the first 6 terms and the 100th term for the sequence defined recursively by the conditions: b1 = 3, and bn = bn-­‐1 + 2 for all n > 1. Limit of a Sequence: Let {an} be a sequence of real numbers, and consider lim!→! 𝑎! . If the limit is a finite number L, the sequence converges and L is the limit of the sequence. IF the limit is infinite or nonexistent, the sequence diverges. Determine whether the sequence converges or diverges. If it converges give the limit: a) 1/1, 1/2, 1/3, 1/4,…,1/n,… b) 2/1, 3/2, 4/3, 5/4, … c) 2, 4, 6, 8, 10, …. d) -­‐1, 1, -­‐1, 1, -­‐1, …, (-­‐1)n, … !!
e) !!! f)
!!!
!! !!
Arithmetic Sequence: A sequence {an} is an arithmetic sequence it can be written in the form {a, a + d, a + 2d, … , a + (n-­‐1)d, … } for some constant d. The number d is called the common difference. Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d: an = an-­‐1 + d (for all n > 2). For each of the following arithmetic sequences find the common difference, the tenth term, a recursive rule for the nth term, and an explicit rule for the nth term. a) -­‐6, -­‐2, 2, 6, 10, … b) ln 3, ln 6, ln 12, ln 24,… Geometric Sequence: A sequence {an} is a geometric sequence if it can be written in the form {a, a∙ 𝑟, 𝑎 ∙ 𝑟 ! , … , 𝑎 ∙ 𝑟 !!! , ….} for some nonzero constant r. The number r is called the common ratio. Each term in a geometric sequence can be obtained recursively from its preceding term by multiplying by r: an = an-­‐1 ∙ r (for all n > 2). For each of the following geometric sequences, find the common ratio, the tenth term, a recursive rule for the nth term, and an explicit rule for the nth term: a) 3, 6, 12, 24, 48, … b) 10-­‐3, 10-­‐1, 10, 103, 105, … The second and fifth terms of a sequence are 3 and 24 respectively. Find the explicit and recursive formulas for the sequence if it is arithmetic and geometric. Graph the sequence {ak} in which ak = k2 – 1. Assignment page 739 # 3-­‐ 24 (3s) page 728 # 27 page 715 #30 page 708 # 33 9.5 Series Sum of a Finite Arithmetic Sequence: Let {a1, a2, a3, …, an} be a finite arithmetic sequence with common difference d. Then the sum of the terms of the sequence is !
𝑎! = 𝑎! + 𝑎! + ⋯ + 𝑎! !!!
𝑎! + 𝑎!
) 2
𝑛
= (2𝑎! + (𝑛 − 1)𝑑) 2
A corner section of a stadium has 8 seats along the front row. Each successive row has two more seats than the row preceding it. If the top row has 24 seats, how many seats are in the entire section? Sum of a Finite Geometric Series: Let {a1, a2, a3, …. , an} be a finite geometric sequence with common ration r ≠ 1. Then the sum of the terms of the sequence is = 𝑛 (
!
𝑎! = 𝑎! + 𝑎! + ⋯ + 𝑎! !!!
𝑎! (1 − 𝑟 ! )
= 1−𝑟
Find the sum of the geometric sequence 4, -­‐4/3, 4/9, -­‐4/27, … , 4(-­‐1/3)10. Infinite Series: An infinite series is an expression of the form: !
𝑎! = 𝑎! + 𝑎! + ⋯ + 𝑎! + ⋯ !!!
For each of the following series, find the first five terms in the sequence of partial sums. Which of the series appears to converge: a) 0.1 + 0.01 + 0.001 + 0.0001 + … b) 10 + 20 + 30 + 40 + … !!!
Sum of an Infinite Geometric Series: The geometric series !
converges if !!! 𝑎 ∙ 𝑟
and only if |r| < 1. IF it does converge, the sum is a/(1 – r). Determine whether the series converges. If it converges, give you the sum. !
4
(− )! 5
!!!
1+ ½ + ¼ + 1/8 + …. Express 0.234234234234… in fraction form. Assignment page 749 # 3-­‐24 (3s) page 739 # 27 page 728 # 30 page 715 #33 page 708 # 36 9.6 Mathematical Induction Let Pn be a statement about the integer n. Then Pn is true for all positive integers n provided the following conditions are satisfied: 1. P1 is true (the anchor) 2. If Pk is true then Pk+1 is true. (the inductive step) We reason by induction when we use evidence derived from particular examples to draw conclusions about general principles. Prove that 1 + 3 + 5 + … + (2n -­‐1) = n2 is true for all positive integers n. !(!!!)(!!!!)
Prove that 12 + 22 + 32 + … + n2 = is true for all positive integers n. !
Prove that 4n – 1 is evenly divisible by 3 for all positive integers n. Assignment page 756 #3-­‐24(3s) page 749 # 27 page 739 # 30 page 728 # 33 page 715 #36 page 708 # 39 9.7 Statistics and Data (Graphical) The table below gives the percentage of each state’s population that was 65 or older in the last official Census. Make a stem plot of this data: AL 13 HI 13.3 MA 13.5 NM 11.7 SD 14.3 AK 5.7 ID 11.3 MI 12.3 NY 12.9 TN 12.4 AZ 13 IL 12.1 MN 12.1 NC 12 TX 9.9 AR 14 IN 12.4 MS 12.1 ND 14.7 UT 8.5 CA 10.6 IO 14.9 MO 13.5 OH 13.3 VT 12.7 CO 9.7 KS 13.3 MT 13.4 OK 13.2 VA 11.2 CT 13.8 KY 12.5 NE 13.6 OR 12.8 WA 11.2 DE 13 LA 11.6 NV 11 PA 15.6 WV 15.3 FL 17.6 ME 14.4 NH 12 RI 14.5 WI 13.1 GA 9.6 MD 11.3 NJ 13.2 SC 12.1 WY 11.7 We can create different types of stem plots. A split-­‐stem stem plot spreads the data out a little bit, putting leaves 0-­‐4 on the lower stem and 5-­‐9 on the upper stem. Back to back stem plots use the same stems, but the leaves from one set of data are added to the left, while leaves from another set are added to the right. Frequency Tables: the number of leaves for a particular stem is the frequency of observations within each stem interval and are often recorded in a table. A table shows the frequency distribution –literally the way that the total frequency is “distributed” among the various home run intervals. A histogram, closely related to a stemplot, displays the information of a frequency table. Make a histogram of Mark McGwire’s Yearly Home Run Totals. Home Runs Frequency Home Run Frequency 0-­‐9 3 40-­‐49 2 10-­‐19 0 50-­‐59 2 20-­‐29 2 60-­‐69 1 30-­‐39 5 70-­‐79 1 Total: 16 If we can make a scatter plot of the data against the time that it was measured, we can analyze the patterns as the variable changes over time. To help with this visualization, the discrete points from left to right are connected by line segments, just as a grapher would do in connect mode. The resulting line graph is called a time plot. To help visually compare data, you can plot two time plots on the same axis (overlaying two time plots). The table below gives the number of cassette units shipped to retailers each year in the 13-­‐year period from 1991-­‐2003. Yea 199 92 93 94 95 96 97 98 99 200 0 02 03 r 1 0 1 Cas 360. 366. 339. 345. 272. 225. 172. 158. 123. 76 4 31. 17.
s 1 4 5 4 6 3 6 5 6 5 1 2 Assignment page 768 # 3-­‐24 (3s) page 756 #27 page 749 # 30 page 739 # 33 page 728 # 36 page 715 #39 page 708 # 42 9.8 Statistics and Data (Algebraic) A 1996 study called Kids These Days: What Americans Really Think About the Next Generation reported that 33% of adolescents say there is no adult at home when they return from school. The report was based on a survey of 600 randomly selected people aged 12-­‐17 years old and had a margin of error of 4%. Did the survey measure a parameter or a statistics, and what does “margin of error” mean? ! !! !⋯!!!
The mean of a list of n numbers is ! !!
, what most people usually think of as “average”. The median of a list of n numbers (arranged in order – either ascending or descending) is the middle number if n is od, and the mean of the two middle numbers if n is even. The mode of a list of numbers is the number that appears must frequently in the lint. Find the mean, median and mode for the following data: Score 10 9 8 7 6 5 4 3 2 1 0 Frequency 2 2 3 8 4 3 3 2 1 1 1 At Marty’s school, it is an administrative policy that the final exam must count 25% of the semester grade. If Marty has an 88.5 average going into the final exam, what is the minimum exam score needed to earn a 90 for the semester? The range of a set of data is the difference between the maximum and the minimum. Another way to measure data is the interquartile range, which is the range o the middle half of the data. Just as the median separates the data into halves, the quartiles separate the data into fourths. The first quartile, Q1 is the median of the lower half of the data, the second quartile is the median, and the third quartile Q3 is the median of the upper half of data. The interquartile range (IQR) measures the spread between the first and third quartiles, comprising the middle half of the data: IQR = Q3 – Q1. The Five-­‐Number Summary: The five-­‐number summary of a data set is the collection: {minimum, Q1, median, Q3, maximum} Find the five-­‐number summaries for the male and female life expectancies in South American nations and compare the spreads. Nation Male Female Argentina 72 79.7 Bolivia 62.5 67.9 Brazil 67.5 75.6 Chile 73.1 79.8 Colombia 67.6 75.4 Ecuador 73.2 79 Guyana 60.1 64.8 Paraguay 72.1 77.3 Peru 67.5 71 Suriname 66.8 71.6 Uruguay 72.7 79.2 Venezuela 71 77.3 A boxplot (sometimes called a box-­‐and-­‐whisker plot) is a graph that depicts the five number summary of a data set. The plot contains a central rectangle (box) that extends from the first quartile to the third quartile, with a vertical segment marking the median. Line segments (whiskers) extend at the ends of the box to the minimum and maximum values. Draw a box and whisker plot for the data from the last example. A number in a data set can be considered an outlier if it is more than 1.5 x IQR below the first quartile or above the third quartile. Is 61 an outlier for the following five number summary for data? {5, 11, 19.5, 30.5, 61} The standard deviation of the numbers {x1, x2, …., xn} is 𝜎=
1
𝑛
!
(𝑥! − 𝑥)! !!!
where 𝑥 denotes the mean. The variance is 𝜎2 the square of the standard deviation. A researcher measured 30 newly hatched loon chicks and recorded their weights in grams. Based on the sample, estimate the mean and standard deviation for the weights of newly hatched loo chicks. Are these measures useful in this case or should we use the five-­‐number summary?: 79.5 87.5 88.5 89.2 91.6 84.5 82.1 82.3 85.7 89.8 84 84.8 88.2 88.2 82.9 89.8 89.2 94.1 88 91.1 91.8 87 87.7 88 85.4 94.4 91.3 86.4 85.7 86 Although we use the word normal in many contexts to suggest typical behavior, in the context of statistics and data distributions it is really a technical term. If you !! !
! graph the function y = 𝑒
The shape corresponds to the kind of distribution we have been calling “normal” – or a Gaussian curve.. The 68-­‐95-­‐99.7 Rule: If the data for a population are normally distributed with mean 𝜇 and standard deviation 𝜎, then • Approximately 68% of the data lie between 𝜇 − 𝜎 and 𝜇 + 𝜎 • Approximately 95% of the data lie between 𝜇 − 2𝜎 and 𝜇 + 2𝜎 • Approximately 99.7% of the data lie between 𝜇 −3𝜎 and 𝜇 + 3𝜎 Based on the research data presented in the last example, would a loon chick weighing 95 grams be in the top of 2.5% of all newly hatched loon chicks? Assignment page 762 # 3-­‐24(3s) page 768 # 27 page 756 #30 page 749 # 33 page 739 # 36 page 728 # 39 page 715 #42 page 708 # 45 Chapter 9 Review page786 # 5 – 110 (5s)