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by: Miss Johan Jacobson


Marketplace > Oregon State University > Mathematics (M) > MTH 245 > MATHEMATICS FOR MANAGEMENT LIFE AND SOCIAL SCIENCES
Miss Johan Jacobson
GPA 3.6


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Class Notes
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This 15 page Class Notes was uploaded by Miss Johan Jacobson on Monday October 19, 2015. The Class Notes belongs to MTH 245 at Oregon State University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/224456/mth-245-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15
Warmug Problem 63 64 An of ce hires 3 workers from 6 applicants In how many different ways can this be done Math 245 Section 65 o A combination is a subset ofdistinct elements selected from a set with no regard to order 0 3 Keys for when it is correct to count combinations Chosen from a single set No repetition No ordering of selected elements Ex1 List all combinations of2 elements from the set a b c Combination counting formula CW n 351k Shortcuts for special cases Cn1 n Cn0 1 Cnn 1 Ex1 An of ce hires 3 workers from 6 applicants ln how many different ways can this be done Ex2 From a set of math books 9 science books and 5 literature books how many different ways can a student pick 2 of each subject Ex3 How many different committees can be formed from a group of 9 women and 11 men if a committee is made of 3 men or 3 women Ex4 A scout troup has 22 members A 5 member team has a leader a cook and a cleanup crew of 3 How many different teams are possible Ex5 A club has 9 seniors 16 juniors 14 sophomores and 10 freshmen Two representatives will be chosen which must be either a senior and a freshmen or a sophomore and a junior How many different choices are possible Ex6 In how many different ways can Ann select one or more of8 friends to invite to a party Warmup Problem 6162 Suppose A B and C are subsets of U We are told nA 15 nB 12 nC 15 nAmB6 nAmC5 nCnB3 nAmBmC2 nA 20 Draw a Venn diagram to represent this situation Math 245 Section 63 c We need some formulas for counting the number of elements in a set 0 These type of counting problems are often worded as counting the number of different ways an activity or a sequence of activities can be done Ex1 Suppose Susan always selects one item from each category in the cafeteria The categories are A orange juice or tomato juice B cereal or eggs and C toast or muf n How many different meals are possible Multiplication Rule Suppose you have activities A1A2 Ak that can be performed in quotpnzamank different ways respectively The number of different ways in which you can perform A1 and A2 and and Ak is found by calculating n1 n2 39 39 quotk Ex2 A manufacturer of computer disks packs them 10 to a box lfwe know a box has 4 defective disks in how many different ways can we select one good disk and one defective disk Ex3 Bob will eat either a bagel or a muf n There are 5 kinds of bagels and 6 kinds of muf ns to choose from How many different choices does Bob have Addition Rule Suppose you have activities A1A2 Ak that can be performed in quotpnzamank different ways respectively The number of different ways in which you can perform A1 or A2 or or Ak is found by calculating n1n2mnki Ex4 In how many different ways can 3 men and 3 women be seated in a row if no one sits next to someone of the same gender Ex5 n how many different ways can 5 art books and 3 music books be arranged on a shefif books of the same subject are kept together Ex6 A 3digit number is formed using digits from the set 1 2 3 4 5 a How many different numbers are there if digits cannot be repeated b What ifdigits can be repeated c How many even numbers with no repeated digits d How many multiples of 5 if digits can be repeated Warm up 21 A bagel shop has tables that seat 2 people and tables that seat 4 people There are 20 tables and 66 people can be seated How many tables of each size are there Math 245 Section 22 Solving any size system of linear equations using an elimination method 0 A matrix plural matrices is a rectangular array ofindividual numbers Example o The augmented matrix of a system of equations is a matrix containing the coef cients and constants of each equation in the system Each row represents an equation Each column represents a certain variable or a constant amount We use this matrix to organize our work when solving with elimination Ex1 a Create the augmented matrix for the system of equations from the warm up problem b Create the augmented matrix for the solutions from the warm up problem Next we will learn how to get from matrix a to matrix b using a series of elimination steps Each equation operation we could do in Section 21 now becomes a row operation Allowable operations 0 Multiply a row by any nonzero constant 0 Add 2 rows and record in one row of the matrix 0 Change the order ofthe rows in the matrix This method of solving systems of equations is called the GaussJordan Method of elimination Ex2 Solve the system of equations 2x 4y 62 22 3x8y5227 xy222 Ex3 Solve the system of equations x2yz 3 y z 1 y52 14 Warmup Problem 72 Two people are selected at random from a group of 12 Republicans and 10 Democrats a Find the probability both are Democrats b Find the probability one is a Republican and one is a Democrat Math 245 Section 73 Compound Events 0 An event is a subset of the sample space 0 Compound events are formed from one or more other events using set operations Given E and F events in S E UFI E OF E Formula for PE39 PE PE PS 1 So PE1 PE and PE1 PE Ex1 Suppose the probability of nding a parking spot at a certain lot at 9 AM is 60 What is the probability you do not nd a spot Ex2 Three equal prizes will be given in a drawing that 50 people entered f4 friends were among the 50 who entered what is the probability at least one of the friends will win a pnze Formula for PEUF HEW PE UF PE PF PE nF Ex3 Studies on a new prescription drug nd it causes headaches in 3 of users drowsiness in 35 and both in 15 Suppose a user is selected at random a What is the probability this user has at least one side effect b What is the probability the user has neither side effect De nition If PE F 0 we say E and F are mutually exclusive events Ex4 Mutually exclusive or not a E drawing a King from a deck of cards F drawing a face card from a deck of cards b E drawing a spade from a deck of cards F drawing a red suit card from a deck of cards Ex5 Four cards are drawn at random from a standard deck a What is the probability all 4 are aces b What is the probability all 4 are jacks c What is the probability all 4 are aces or all 4 are jacks Ex 6 Three numbers are drawn from 1 234567 and placed in the order drawn to form a 3digit number with no repeated digits a What is the probability the number is 456 b What is the probability the number is larger than 600


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