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PROJECT 5: PROBABILITY
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%.
(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.