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Question;The Problems;Misconceptions about how probability works are common. The following questions;will require you to apply your mathematical reasoning skills to avoid these;common mistakes.;Answer the following;questions. As always, justify your answers with work or an explanation.;(1) On;his way to work every day, Maxwell passes two traffic lights. After many;commutes, he notices that there seems to be around a 50% chance of being;stopped at the first light. He also notices that there does not seem to be any;relation between the two lights. Regardless of whether or not he must stop at;the first, the probability that he is stopped at the second light is also;around 50%.;Assume;Maxwell's probability estimates are accurate. During a commute, Maxwell could;be stopped twice, once, or zero times. Since there is a 50% chance of being;stopped at each light, Maxwell decides that the?probability that he will be;stopped exactly once during a given commute is 1/3 and the probability that he;will be stopped twice is 1/3. He concludes that the probability of being;stopped at least once during a given commute is 2/3. Both Maxwell's reasoning and answer are wrong!Calculate the;correct probability that Maxwell will be stopped at least once during a given;commute.;(2) Suppose;you flip a fair coin four times in a row. Consider the following two possible;outcomes.?;Flip;number 1st 2nd 3rd 4th;Side;H H H H?;Flip;number 1st 2nd 3rd 4th;Side;H T T H?;Lawrence;believes that the first outcome (flipping 4 heads in a row) is less likely to;occur than the second outcome. Is Lawrence right or wrong? To answer the;question, calculate the probability of each of these two outcomes.;(3) Imagine;the following game. You are offered two hats that are filled with marbles all;of the same size. Hat A contains 25 marbles, exactly 10 of which are red. Hat B;contains exactly 30 red marbles. In this game, you first pick either hat A or;hat B. You then choose one marble at random from the hat that you picked. If you;choose a red marble, then you win. Your goal is to determine which hat you;should pick to maximize your probability of winning.;(a);Explain;why you do not have enough information to decide which hat to pick. What;additional data do you need to know?;(b);Hat B;contains 30 red marbles while hat A only contains 10, so many people would;decide to play this game with hat B without further thought. However, that is;not always the best choice. Think back to the additional information that you;decided was needed in part (a). Give an example of a specific value for this;extra data that would cause you to be more likely to win by picking hat A.;(4) A;one-hundred-year flood is a flood;that is so severe that it is only expected to occur on average once every 100 years. More formally, in a given year, the probability of a one-hundred-year flood;occurring is 1/100 =1%.;(a);A certain;river has not reached its one-hundred-year flood level in the past 99 years.;What is the probability that it will reach its one- hundred-year flood level;during the next year?;(b) A different river reached its;one-hundred-year flood level last year. What is the probability that this river;will reach its one-hundred-year flood level again during this year?;(5) Every;year, approximately 100 drivers hit a deer in Hawaii.;(a) The current U.S. population is;approximately 319 million. Compute the probability that a randomly chosen;American will have crashed?into a deer in Hawaii in the past year.;(b) Consider the probability that;you found in part (a). Does this give?the probability that you will hit a deer;in Hawaii during the next year? Do you expect your probability of hitting a;deer in Hawaii to be greater than, less than, or the same as the probability;that you computed in part (a) Justify your answers with an explanation.


Paper#60223 | Written in 18-Jul-2015

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