Chapter 8: Further Topics in Algebra

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## Chapter 8: Further Topics in Algebra

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**Chapter 8: Further Topics in Algebra**8.1 Sequences and Series 8.2 Arithmetic Sequences and Series 8.3 Geometric Sequences and Series 8.4 The Binomial Theorem 8.5 Mathematical Induction 8.6 Counting Theory 8.7 Probability**8.7 Probability**Basic Concepts • An experiment has one or more outcomes. The outcome of rolling a die is a number from 1 to 6. • The sample space is the set of all possible outcomes for an experiment. The sample space for a dice roll is {1, 2, 3, 4, 5, 6}. • Any subset of the sample space is called an event. The event of rolling an even number with one roll of a die is {2, 4, 6}.**8.7 Probability**Probability of an Event E In a sample space with equally likely outcomes, the probability of an event E, written P(E), is the ratio of the number of outcomes in sample space S that belong to E, n(E), to the total number of outcomes in sample space S, n(S). That is,**8.7 Finding Probabilities of Events**Example A single die is rolled. Give the probability of each event. (a) E3 : the number showing is even (b) E4 : the number showing is greater than 4 (c) E5 : the number showing is less than 7 (d) E6 : the number showing is 7**8.7 Finding Probabilities of Events**Solution The sample space S is {1, 2, 3, 4, 5, 6} so n(S) = 6. (a) E3 = {2, 4, 6} so (b) E4= {5, 6} so**8.7 Finding Probabilities of Events**Solution (c) E5 = {1, 2, 3, 4, 5, 6} so (b) E6 = Ø so**8.7 Probability**• For an event E, P(E) is between 0 and 1 inclusive. • An event that is certain to occur always has probability 1. • The probability of an impossible event is always 0.**8.7 Complements and Venn Diagrams**• The set of all outcomes in a sample space that do not belong to event E is called the complement of E, written E´. If S = {1, 2, 3, 4, 5, 6} and E = {2, 4, 6} then E´ = {1, 3, 5}.**8.7 Complements and Venn Diagrams**• Probability concepts can be illustrated with Venn diagrams. The rectangle represents the sample space in an experiment. The area inside the circle represents event E; and the area inside the rectangle but outside the circle, represents event E´.**8.7 Using the Complement**Example A card is drawn from a well-shuffled deck, find the probability of event E, the card is an ace, and event E´. Solution There are 4 aces in the deck of 52 cards and 48 cards that are not aces. Therefore**8.7 Odds**The odds in favor of an event E are expressed as the ratio of P(E) to P(E´) or as the fraction**8.7 Finding Odds in Favor of an Event**Example A shirt is selected at random from a dark closet containing 6 blue shirts and 4 shirts that are not blue. Find the odds in favor of a blue shirt being selected. SolutionE is the event “blue shirt is selected”.**8.7 Finding Odds in Favor of an Event**Solution The odds in favor of a blue shirt are or 3 to 2.**8.7 Probability**Probability of the Union of Two Events For any events E and F,**8.7 Finding Probabilities of Unions**Example One card is drawn from a well-shuffled deck of 52 cards. What is the probability of each event? (a) The card is an ace or a spade. (b) The card is a 3 or a king.**8.7 Finding Probabilities of Unions**Solution (a) P(ace or space) = P(ace) + P(spade) – P(ace and spade) (b) P(3 or K) = P(3) + P(K) – P(3 and K)**8.7 Probability**Properties of Probability 1. 2. P(a certain event) = 1; 3. P(an impossible event) = 0; 4. 5.**8.7 Binomial Probability**An experiment that consists of • repeated independent trials, • only two outcomes, success and failure, in each trial, is called a binomial experiment.**8.7 Binomial Probability**Let the probability of success in one trial be p. Then the probability of failure is 1 – p. The probability of r successes in n trials is given by**8.7 Finding Binomial Probabilities**Example An experiment consists of rolling a die 10 times. Find the probability that exactly 4 tosses result in a 3. Solution Here , n = 10 and r = 4. The required probability is