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Math 101 – Exam 3 Review (Chapters 2 and 3 - concepts from 2.1 are listed but will be less emphasized, since they were covered on Exam 2) Historical Figures (* are most important) *John Venn (again) *Augustus DeMorgan *Blaise Pascal *Pierre de Fermat *Gregor Mendel *Reginald Punnett *Nancy Wexler Christian Kramp Antoine Gombauld Math Concepts Sets and their properties o Empty sets, the Universal set, and mutually exclusive sets o Union and Intersection o Cardinal Number formula (related to union and intersection) o The Complement of a Set Venn Diagram Shading Blood Types DeMorgan’s Law for simplifying complements 2-circle and 3-circle Venn Diagrams – cardinal numbers of subsets and probability calculations Combinatorics – 3 main problem types: o Counting principles for independent events o Permutations o Combinations Factorial notation, and reducing factorial fractions Distinguishable permutations Probability vocabulary o experiments o outcomes o sample space o event o odds - true odds vs. house odds o Probability (aka theoretical probability) vs. relative frequency (aka experimental probability) Coin flipping calculations – sample space, probabilities of events Law of large numbers Mendel’s laws of genetics (dominant and recessive genes) Punnett Squares Inherited diseases: o Cystic Fibrosis & Tay Sachs (recessive) o Sickle Cell Anemia (co-dominant) o Huntington’s (dominant) Range of possible probabilities; probabilities of certain and impossible events Mutually Exclusive events and their probabilities Roulette (if a problem of this type is given, a chart of the house odds will be provided) Pair of Dice calculations - sample space, probabilities of events Complements in probability calculations (e.g., the paintball problem, the birthday problem). Probabilities in Card Hands Lottery Calculations Expected Value of experiments o games - roulette, lottery, Keno, cards o other “risk-value” problems, like insurance Decision Theory - statistical and psychological factors Applications - There are many applications scattered throughout the math concepts, including genetics, diseases, roulette, lotteries, insurance Practice Problems 1. How many 3-digit house numbers can be formed from the digits 1, 2, 3, 4, 5, 6, 7 if the first digit must be 1, but replacement is allowed (digits may be reused)? 2. a) Find the value: 12 P3 b) Find the value and simplify: x P1 3. For a standard deck of cards with no jokers, Let event Q = “drawing a queen”, event R = “drawing a red card” a) Find p(Q R) b) Find o(Q) 4. The gene for Tay-Sachs disease (t) is recessive, and (N) represents a normal gene. If 2 parents have genotypes types NN and Nt: a) What is the probability that their child will have Tay-Sachs disease? b) What is the probability that their child will be a carrier? c) What is the probability that their child will neither have the disease, nor be a carrier? 5. The probabilities of amounts a customer will spend on a pair of shoes are shown below. What is the expected value of the amount that a customer will spend on shoes? Amount Probability 0 0.3 $30 0.4 $50 0.2 $80 0.1 6. At a DMV (Dept. of Motor Vehicles office), the following probabilities are observed: Prob. of passing the written test on the first try: 0.45 Prob. of passing the road test on the first try: 0.35 Prob. of passing at least one test on the first try: 0.7 a) Find the probability of failing both tests on the first try b) Find the probability of passing both tests on the first try 7. Of 40 students surveyed, the # of students who completed each class is listed: 19 25 26 5 16 6 2 a) b) c) d) Art Music PE (Physical Education) completed all 3 Music and PE Music only Art only Fill in the Venn diagram showing appropriate numbers for each subset. How many respondents did not complete any of the 3 classes? What percent of all students took Music? What percent of all students took Art or PE?