Math 123 Finals Mass Tutorial

Math 123 Finals Mass Tutorial - MATH 123 MASS TUTORIAL W E...

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Unformatted text preview: MATH 123 MASS TUTORIAL W E D N E S D A Y A P R I L 6 T H , 2 0 1 1 T H U R S D A Y A P R I L 7 T H , 2 0 1 1 B R O U G H T T O Y O U B Y J U L I E P A R K A N D Y U R O N G F A N Chapter 5 Chapter 5: Mathematics of Finance Variables t time in years P original amount in account (principal) A final amount in account n - # of compounding interests (n = m x t) FV: Future Value PV: Present Value PMT: payment r: interest rate i: r/m # of times interest is compounded = m yearly bi-yearly monthly quarterly weekly daily m=1 m=2 m = 12 m=4 m = 52 m = 365 Formulas Simple Interest: Compound Interest: If an amount P is deposited in an account paying r% compounded m times a year for t years, then the amount in the account after t Continuous Compounding: Practice! #2 If you put $10,000 in a bank account paying a nominal rate of 3.8% and take the money out after 10 years. How much will you earn in interest if Yearly Monthly Continuously Effective Rate Practice! Find the effective rate of interest corresponding to a nominal interest rate of 5.3% compounded Annually Quarterly Monthly Daily Future Value If an amount, PMT (payment), is deposited m times in an account paying r% compounded m times a year for t years, the final amount in the account after t When to use FV? Want to find/given the desired final value after t years Interest works for you Practice! #6 If Tom deposits $100 each month in a savings account earning interest at a nominal rate of 7.4% per year compounded monthly, how much will he have on deposit in his savings account after 4 years? Practice! #12 A company estimates that it will need $100,000 6 years from now to organize a certain event. The bank pays 2.5% compounded monthly. How much should the company put in the bank account today so it has the needed amount in 6 years? How much interest did the company earn? The company decides to make monthly payments instead of putting the whole sum in at once. How much should each payment be so the company has the needed amount in 6 years? Present Value Borrow PV dollars now (in present) and want to pay all/part of the amount by making a payment PMT m times a year in an account with r% interest rate compounded m times a year, When to use PV? Borrow money now and have to pay over time Problem works against you you pay more than you borrowed (due to interest) Practice! #7 Margaret made a down payment of $3000 towards the purchase of a car. To pay the remaining balance, she has secured a loan from her bank at rate of 6.6% per year compounded monthly. She is required make payments of $150 per month for 48 months. What is the price of the car? How much in total will Margaret spend on interest charges? Practice! #14 The bank offers John a $20,000 loan to finish his 2 years of MBA. One year after the completion of the program, John must pay back the loan over 5 years with monthly deposits. The bank charges him 7.2% compounded monthly How much interest did he pay? After 3 years, the bank offers John the opportunity to pay remaining balance in total. How much is the remaining balance? Chapter 7 Sets Set: well-defined collection of objects/outcomes Cardinality of a set: # of elements in that set Denoted by n(A) C: complement of set A All elements that is NOT in set A, applied when given a universal set Universal set: set containing all possible elements A C B: Subsets A is a subset of B denoted if every element of A is also an element of B All elements of A are in B Given a set A, we define the power set, P(A), to be the set of subsets of A i.e. if A = {a, b, c, d}, then there are 16 possible subsets i.e. {a}, {a,b}, {a,b,c}, {a,b,c,d}, {b},...{0} n(P(A)) = 2n ``A and B`` Intersection between A and B elements in common with both sets (think overlap) ``A or B`` Union of A and B combine the 2 sets (includes ALL elements of both sets) Theorem Practice! #1. If n(A) = 29, n(B) = 18 and n (AnB) = 14 then n (A U B) = #2. If n (A U B) = 39, (A n B) = 5, and n(A) = n (B); then n(A) = Practice! # 3 Find the number of elements in A U B U C if there are 200 elements in A, 1200 elements in B, and 3200 elements in C in each of the following cases The sets are pairwise disjoint. A C B C C There are 20 elements in common to each pair of sets & 7 elements in all three sets. Practice! #4. There are a total of 371 students in a college who have taken a course in calculus, 212 who have taken a course in statistics, and 160 who have taken a course in both calculus and statistics. How many students at this college have taken a course in either calculus or statistics? Probability Sample Space: set containing all possible outcomes Event: a subset of the sample space Probability of an event: E -event S -sample space Odds in favor of an event E occurring: Practice! #8 Consider that there are 365 days a year. If 2 people are chosen, what is the probability of Having the same birthday? Being born in the same month? Practice! #11 A coin is tossed 3 times. Find the probability of Getting at least one head Getting at least two heads Getting an even number of heads Conditional Probability Conditional Probability: probability of an event E occurring given event F has already occurred Essentially reduces sample space to F Independent Events E & F are independent if Proof: occurrence of F does not affect probability of E Mutually exclusive events: two events are mutually exclusive if they have no simple events in common cannot happen at the same time Practice! Practice! #12 In a store there is sweet and salty food. Consumer data shows that 8 out of 20 customers will buy sweets, 6 will buy salty food, and 2 will buy both. What is the probability that A customer buys nothing A customer buys only sweets A customer buys sweets knowing he already bought salty food. Practice! #16 What is the probability that when 2 fair dice are tossed, at least one of them is 6 if we know that the sum is 8? Practice! #17 Urn A contains 5 red balls and 7 blue balls. Urn B contains 4 red balls and 3 blue balls. We throw a die: if the outcome is even, we pick a ball from A; if the outcome is odd, we pick a ball from B. Suppose a red ball is selected, what is the probability that the outcome was even? Theorem Inverse probability: knowing the conditional probability of A given B, what is the conditional probability of B given A Practice! #19 Suppose there is a school with 60% boys and 40% girls as its students. The female students wear trousers or skirts in equal numbers; the boys all wear trousers. An observer sees a (random) student from a distance, and what the observer can see is that this student is wearing trousers. What is the probability this student is a girl? Chapter 8 Outline Permutation Combination Probability with Counting Binomial Probability Probability Distributions Factorial 0! = 1 Example: If there are 4 books on shelf, how many ways can they be arranged? 4 * 3 * 2 * 1 = 4! = 24 Permutations Arranging objects into a particular order Order matters! n = number of elements in total r = number of choices to be made Permutation Example Eight candidates sought the Democratic nomination for president. In how many ways could voters rank their first, second, and third choices? 2. Number of permutations of letters A, B, and C 3. How many different 4-letter radio station call letters can be made if: 1. a) b) c) first letter must be K or W and no letter may be repeated? Repeats are allowed, but the first letter is K or W? First letter is K or W, there are no repeats, and the last letter is R? Permutation Example 4. There are 3 pyramids, 4 cube, and 7 spheres. a) b) c) d) In how many ways can the objects be arranged in a row if each is a different color? How many arrangements are possible if objects of the same shape must be grouped together and each object is a different color? In how many distinguishable ways can the objects be arranged in a row if objects of the same shape are also the same colour, but need not be grouped together? In how many ways can you select 3 objects, one of each shape, if the order in which the objects are selected does not matter and each object is a different color? Combination Arranging without regard to order Order does not matter! n= number of elements in total r= number of choices to be made Combination Example 1. 3 lawyers are to be selected from a group of 30 to work on a special project. a) b) In how many different ways can the lawyers be selected? In how many ways can the group of 3 be selected if a certain lawyer must work on the project? How many such hands have only face cards? How many such hands have exactly 2 hearts? How many such hands have cards of a single suit? 2. Five cards are dealt from a standard 52-card deck a) b) c) Combination Example 3. Hamburger Hut sells regular hamburgers as well as a larger burger. Either type can include cheese, relish, lettuce, tomato, mustard, or catsup. a) How many different hamburgers can be ordered with exactly three extras? b) How many different regular hamburgers can be ordered with exactly three extras? c) How many different regular hamburgers can be ordered with at least five extras? Probability with Counting Principles In a common form of card game poker, a hand of 5 cards is dealt to each player from a deck of 52 cards. Find the probability of getting each of the following hands: a) A hand containing only hearts b) A flush of any suit (5 cards of the same suit) c) A full house of aces and eights (3 aces and 2 eights) 1. Example When shipping diesel engines abroad, it is common to pack 12 engines in one container that is then loaded on a rail car and sent to a port. Suppose that a company has received complaints from its customers that many of the engines arrive in nonworking condition. To help solve this problem, the company decides to make a spot check of containers after loading. The company will test 3 engines from a container at random; if any of the 3 are nonworking, the container will be shipped until each engine in it is checked. Suppose a given container has 2 nonworking engines. Find the probability that the container will not be shipped. Binomial Probability Rules: The same experiment is repeated a fixed number of times There are only two possible outcomes, success and failure The repeated trials are independent, so that the probability of success remains the same for each trial n = # of independent repeated trials of experiment x = # successes p = probability of sucess Binomial Probability Example Find the probability of getting exactly 7 heads in 8 tosses of a fair coin 2. A recent study found that 85% of breast-cancer cases are detectable by mammogram. Suppose a random sample of 15 women with breast cancer are given mammograms. Find the probability of each of the following results, assuming that detection in the cases is independent. 1. a) b) c) All of the cases are detectable None of the cases are detectable Not all cases are detectable Probability Distributions Is this a fair game? Expected value of 0 = fair game For binomial probability, E(x) = np Probability Distributions Example 1. a) Suppose a die is rolled 4 times. Find the probability distribution for the number of times 1 is rolled What is the expected number of times 1 is rolled? b) 2. From a group of 3 women and 5 men, a delegation of 2 is selected. Find the expected number of women in the delegation. Chapter 9 Statistics Median: middle entry in a set of data arranged in increasing/decreasing order If there is an even number of entries, the median is defined to be the mean of the 2 center entries Mode: most frequent entry Random sample: chosen so that every element of population is equally likely to be selected representative of the population Mean Practice! #2 Find the mean for the data shown in the following frequency distribution. Value 30 32 33 37 42 Frequency 6 9 7 12 6 #3 Find the mean from the grouped frequency distribution. Interval 0-4 5-9 10-14 15-19 20-24 25-29 30-34 Frequency 3 4 6 8 5 3 1 Variation Standard Deviation Practice! #4 Find the standard deviation of the following numbers: 7, 9, 18, 22, 27, 29, 32, 40 #5 Find the standard deviation for the grouped data. Interval 0-4 5-9 10-14 15-19 20-24 25-29 30-34 Frequency 3 4 6 8 5 3 1 Chapters 2-4 Types of Solutions Solution Inconsistent system no solution Parametric solution Inconsistent solution Solve for x and y given: 9x-5y=1 and -18x+10y=1 9x 5y=1 0x + 0y 3 Parametric solution Solve for x and y given: 2x+y=1 and 6x+3y=3 x+0.5y=0.5 x=0.5(1 y) if y=t where x=0.5(1-t) Word Problem for Row Reducing Nadir Inc. produces three models of television sets: deluxe, super-deluxe, and ultra. Each deluxe set requires 2 hours of electronics work, 2 hours of assembly time, and 1 hour of finishing time. Each super-deluxe requires 1, 3, and 1 hour of electronics, assembly, and finishing time, respectively. Each ultra requires 3, 2, and 2 hours of the same work, respectively. There are 100 hours available for electronics, 100 hours available for assembly, and 65 hours available for finishing per week. How many of each model should be produced each week if all available time is to be used? Rules Addition and Subtraction When adding A and B, they need to be of the same size Multiplication only allowed to multiply if A is m x n, and B is n x k IDENTITY MATRIX Rules: n x n; entries are all 0 except the main diagonal, where the entries are 1 = Inverse A x A-1 = A-1 x A = I A: matrix A-1: inverse of matrix A I: identity matrix [A|I] A-1] Inverse Example Find inverse of Inverse Formula for calculating inverse of 2x2 matrix: *memorize this formula!! Inverse Example Find inverse of: Input-Output Model Set up your input-output matrix output Set up your demand matrix Use formula: X AX = D X = (I A)-1 D x A D I how much you need to produce how much is internally consumed (your input-output matrix) demand function identity matrix Input-Output Model Example An economy consists of 2 sectors: agriculture (A) and manufacturing (M). The production of one unit of agricultural products requires the consumption of 0.1 units of agricultural products and 0.3 units of manufacturing. The production of 1 unit of manufacturing requires the consumption of 0.2 units of agricultural products and 0.4 units of manufacturing. Calculate the gross output of goods needed to satisfy a demand of 10million units of agricultural products & 20million units of manufactured goods Input-Output Model Setting up the Matrix The production of one unit of agricultural products requires the consumption of 0.1 units of agricultural products and 0.3 units of manufacturing. A M A M The production of 1 unit of manufacturing requires the consumption of 0.2 units of agricultural products and 0.4 units of manufacturing A M A M demand of 10million units of agricultural products & 20million units of manufactured goods * in millions of units Solution *X = (I A)-1 D *in millions of units Input-Output Model Summary: Set up your input/output matrix & demand matrix 1. 2. 3. Subtract input/output matrix from the identity matrix Find its inverse Multiple it by the demand function Answer Input-Output Model (Closed) Formula: (I A)X = 0 Demand = 0 Occurs when all columns add up to 1 Matrix is not invertible parametric solution infinitely many solution Represents a ratio Example! Find the ratios of products A, B, C using a closed model A B C Solution: R1 + R2 + R3 -10R2 -10R3 R3 6R2 R3 R1 R2 R2 R3 R1 R1 + 5R2 R1 6 1/33R3 R3 R2 R3 x = 23/33 t y = 31/33 t z = t Ratio 23: 31: 1 Optimizing Graphically Finding the maximum or minimum: Draw feasible set Find the corner points of the region Evaluate the function on each corner point to find the max/min *if 2 points give the same max/min, then the whole line segment joining them is the solution *if the region is unbounded (towards positive infinity), then there is no maximum, only minimum Example Solve the following linear programming problem graphically: Maximize P = 10x1 + 8x2 Subject to: 4x1+2x2 x1+5x2 x1, x2 Same thing as: y = -2x + 10 y = -1/5x + 12/5 Solution Maximize P = 10x1 + 8x2 Subject to: -2x + 10 -1/5x + 12/5 x1, x2 (0, 10) Corner points: P(0,0) = 0 P(0,12/5) = 96/5 P(5,0) = 50 P(38/9, 14/9) = 164/3 (0, 12/5) (38/9, 14/9) (5, 0) (12,0) OPTIMIZE USING SIMPLEX METHOD Rule: 2) variables are positive anxn Slack variables =used to turn inequalities into linear equations e.g. x1 + x2 > 8 To turn this into linear equation, introduce slack variable, s1 x1 + x2 +s1 = 8 OPTIMIZE USING SIMPLEX METHOD Example 1: Maximize z = 5x1 + 2x2 subject to: 1) Turn these inequalities into linear equations 2x1 + 4x2 +s1 = 15 3x1 + x2 + s2 = 10 2) Make z equation equal to 0 (where z is positive) z 5x1 2x2 = 0 3) Put them into matrix x1 x2 s1 s2 z 2 4 1 0 0 15 ------1st equation 3 1 0 1 0 10 ------2nd equation -5 -2 0 0 1 0 -------z equation 4) Find the most negative number from z equation row 5) Look at the quotients of that column and the last column 15/2 = 7.5 10/3 = 3.3 smaller quotient So, 3 is the pivot number make it 1 -NOTE- If quotients are all negative, there is no maximum 6) REPEAT UNTIL YOU HAVE NO NEGATIVES IN THE LAST ROW 7) State x1, x2, s1, s2, z ...
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