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Sarah Miney invests $10,000 in a certain stock at the beginning of the year. By examining past movements of this stock and consulting with her broker, Sarah estimates that the annual return from this stock, X, is normally distributed with mean 5% and standard deviation 14%. Here X (when expressed as a decimal) is the pro t Sarah receives per dollar invested. It means that on December 31, her $10,000 will have grown to $10; 000(1+X) dollars. Because Sarah is in the 33% tax bracket, she will then have to pay the Internal Revenue Service 33% of her pro t, if she earns a pro t. However, she doesn t have to pay any tax if she loses money. Calculate the probability that Sarah will have to pay the IRS at least $400, and calculate the probability that she won't have to pay any tax. Also, calculate the dollar amount such that Sarah's after-tax pro t is 90% certain to be less than this amount; that is, calculate the 90th percentile of his after-tax profi t.
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