probability

Instructions: Solve the following problems, making sure to include both calculations and brief prose descriptions of your solutions. For problems related to the normal distribution, feel free to use a z-score table or calculator. You may also try computing in Python with scipy.stats. (Optionally, as a special challenge, compute the answers directly from an integral.)

Suppose we would like to survey the residents of a small town (pop. 10,000) for their feelings about money and happiness. Our sample size will be n=100. Let X represent the sample, chosen using the strategies below. For which of these strategies would it be appropriate to assign the uniform distribution (i.e., any set of 100 people is just as likely as any other) to X? If it is appropriate, what probability should we assign to each possible sample? (a) Take the first 100 people who walk by you on the street. (b) Ask the town hall to sort the people by their Social Security number or some other form of identification, and then take the first 100 in the resulting list. (c) Ask the town hall for a set of cards, with each card containing the name of exactly one resident, and with each person appearing on exactly one card. Throw the cards out of a third-story window, then walk outside and pick up the first 100 cards that you find.

Under the same conditions as in the preceding exercise, can you describe a procedure which, if used, would select any group of 100 people from the town with equal probability? Can you describe such a procedure that does not rely on a computer or a calculator?

Acme Corp. offers year-end bonuses that are normally distributed with parameters \mu\:=\:\text{100}\:,\:\sigma\:=\:2 . A human resources consultant advises that employees will be happy when bonuses are between $97 and $105. What fraction of the bonuses are likely to make employees unhappy? Management is not willing to change the mean, but they are willing to reduce the value of \sigma . What value of \sigma will ensure that no more than 1 percent of the bonuses are likely to make employees unhappy?

In the martingale doubling system for gambling, the player doubles his bet each time he loses. Suppose that you are playing roulette in a fair casino, and you bet on red each time. The roulette wheel has 38 slots, 19 red and 19 black. You then win with probability 1/2 each time. Assume that you enter the casino with 100 dollars, start with a 1-dollar bet and employ the martingale system. You stop as soon as you have won one bet, or in the unlikely event that black turns up six times in a row so that you are down 63 dollars and cannot make the required 64-dollar bet. Find your expected winnings under this system of play.

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