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3-35. Election day is less than a month away, and political analysts are focused on the races for two statewide political offices. According to a recent poll, in the race for Governor, 40% of voters support the Republican candidate and 53% of voters support the Democratic candidate. In the race for Attorney General, 61% of voters support the Republican candidate and 37% of voters support the Democrat. Assume that voter preference of candidate for each of the two offices is independent of each other. a. What is the probability that a randomly selected voter supports both the Democratic candidate for Governor and the Democratic candidate for Attorney General? (.53)(.37) = 19.6% b. What is the sample space for all of the possible outcomes in voter support of the candidates for Governor and Attorney General? {RR, RD, RO, DR, DD, DO, OR, OD, OO} c. R (.40) D (.53) O(.07) R(.61) D(.37) O(.02) d. A set of outcomes (a subset of the sample space) is called an event. Which outcomes from the sample space are in the event, supporting “Democratic Governor”? Which outcomes are in the event, supporting “Democratic Attorney General”? Democratic Governor: {DR,DD,DO}, Democratic Attorney General: {RD,DD,OD} e. The intersection of two events A and B is the event consisting of all outcomes that are in both A and B. If you have not done so already, create an area model for this situation. What outcomes are in the intersection of the events “Democratic Governor” and “Democratic Attorney General”? How can this be seen in your area model? Shade your area model to highlight the event “Democratic Governor”. Then shade your area model to highlight the event “Democratic Attorney General”. What do you notice about the intersection of the events? 3.1.4 What are the chances of both events? October 22, 2015 Objectives • CO: SWBAT calculate the probability of a union both through intuitive methods and using the Addition Rule. • LO: SWBAT use mathematical language for calculating probabilities of unions, intersections, and complements of events. 3-36. Darren is a Democrat and is really hoping that a Democratic candidate will win at least one of the offices. a. Darren tells his friend Antonia, “Since the Democratic candidate for Governor is supported by 53% of voters and the Democratic candidate for Attorney General is supported by 37% of voters, that means 90% of voters support the Democratic candidate for Governor or Attorney General.” Antonia says, “Well, if that is true, then 101% of voters support a Republican candidate for Governor or Attorney General.” Is Darren’s thinking correct? Why or why not? It can’t be; he forgot about the overlapping events. b. The union of two events A and B is the event consisting of all outcomes that are either in A or in B or in both events. What outcomes from the sample space are in the union of the events “Democratic Governor” and “Democratic Attorney General”? How can this be seen in your area model? {DR, RD, DD, DO, OD}; It is all the outcomes in row and column for Democrat. c. What is the probability of the union of the two events in part (b)? That is, what is the probability that a randomly selected voter supports the Democratic candidate for Governor or the Democratic candidate for Attorney General? .148 + .1961 + .0259 + .3233 + .0106 = 70.39% d. What is the probability that a randomly selected voter supports a Republican for Governor and a Democrat for Attorney General? 14.8% e. What is the probability that a randomly selected voter supports a Republican for Governor or a Democrat for Attorney General? .244 + .148 + .008 + .1961 + .0259 = 62.2% 3-37. Viola described the following method for calculating the probability in part (e) of problem 3-36:“I shaded all the outcomes for a voter preferring a Republican Governor and all the outcomes for a voter preferring a Democratic Attorney General. But I noticed that there were some outcomes that were shaded twice! So, I just added the original probabilities and subtracted the probability of the overlapping outcomes: 0.40 + 0.37 – 0.148 = 0.622.” a. Does Viola’s answer match your answer for part (e) of problem 336? How does her method compare to the method you used? Yes, in not adding the overlapping one twice we were subtracting it. b. Will Viola’s method always work? Why or why not? Yes, it will always work because the overlapping area cannot be counted twice. • 3-38. Viola’s method of “adding the two probabilities and subtracting the probability of the overlapping event” is called the Addition Rule and can be written: • P(A or B) = P(A) + P(B) – P(A and B) • You have already seen that any event where event A or B occurs is called a union and is said “A union B.” The event where both events A and B occur together is called an intersection. So the Addition Rule can also be written: • P(A union B) = P(A) + P(B) – P(A intersection B) • Use the Addition Rule to calculate the probability that a third-party candidate will be elected for either Governor or Attorney General. Then check your results using another method. .07 + .02 – .0014 = .0886 .0427 + .0259 + .0014 + .008 + .0106 = .0886 3-40. Sometimes it is easier to figure out the probability that something will not happen than the probability that it will. When finding the probability that something will not happen, you are looking at the complement of an event. The complement is the set of all outcomes in the sample space that are not included in the event. Show two ways to solve the problem below, then decide which way you prefer and explain why. a. The marketing department has interviewed people of all different age groups about the BBQ chicken salad, but now they need information about why the other salads are less popular. What is the probability that the next person randomly chosen will not prefer the BBQ chicken salad? P(not BBQ) = 0.1 + 0.1 + 0.2 = 0.4 or P(not BBQ) = 1 – P(BBQ) = 1 – 0.6 = 0.4 b. If the probability of an event A is represented symbolically as P(A), how can you symbolically represent the probability of the complement of event A? 1 – P(A)