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Chapter 6: Q. 88 (page 404)

88. Scrabble In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100tiles. There are 42vowels, 56 consonants, and 2 blank tiles in the bag. Cait chooses her7 tiles and is surprised to discover that all of them are vowels. Can we use a binomial distribution to approximate this probability? Justify your answer

Short Answer

Expert verified

No binomial distribution, because the draws are not independent and the constant probability of success is not constant.

Step by step solution

01

Given information

There are 42vowels, 56consonants, and 2blank tiles in the bag. Cait chooses her 7 tiles.

02

Explanation

A random variable has a binomial distribution if:

  • In terms of variables, there are two possible outcomes.
  • Each draw must be self-contained.
  • There are a set number of elements.
  • Probability of success is always the same.

The reason for this is that after selecting a tile, it is not returned to the bag and so cannot be selected again. As a result, the chances of winning after each tie are not the same.

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Most popular questions from this chapter

84. Lie detectors Refer to Exercise 82. Let Y= the number of people who the lie detector says are telling the truth.
(a) Find P(Y鈮�10). How is this related toP(X鈮�2)? Explain.
(b) Calculate 渭Yand蟽Y. How do they compare with 渭Xand 蟽X? Explain why this makes sense.

18. Life insurance

(a) It would be quite risky for you to insure the life of a 21-year-old friend under the terms of Exercise 14. There is a high probability that your friend would live and you would gain \(1250in premiums. But if he were to die, you would lose almost \)100,000. Explain carefully why selling insurance is not risky for an insurance company that insures many thousands of 21-year-old men.

(b) The risk of an investment is often measured by the standard deviation of the return on the investment. The more variable the return is, the riskier the
investment. We can measure the great risk of insuring a single person鈥檚 life in Exercise 14by computing the standard deviation of the income Y that the insurer will receive. Find 蟽Y using the distribution and mean found in Exercise 14.

53. The Tri - State Pick 3Refer to Exercise 42. Suppose you buy a $1 ticket on each of two different days.
(a) Find the mean and standard deviation of the total payoff. Show your work.
(b) Find the mean and standard deviation of your total winnings. Interpret each of these values in context.

Lie detectors A federal report finds that lie detector tests given to truthful persons have probability about 0.2 of suggesting that the person is deceptive. A company asks 12job applicants about thefts from
previous employers, using a lie detector to assess their truthfulness. Suppose that all 12 answer truthfully. Let X=the number of people who the lie detector says are being deceptive.
(a) Find and interpret 渭X.
(b) Find and interpret 蟽X.

52. Study habits The academic motivation and study habits of female students as a group are better than those of males. The Survey of Study Habits and Attitudes (SSHA)is a psychological test that measures these factors. The distribution of SSHAscores among the women at a college has mean120and standard deviation 28, and the distribution of scores among male students has mean 105and standard deviation 35.You select a single male student and a single female student at random and give them theSSHA test.
(a) Explain why it is reasonable to assume that the scores of the two students are independent.
(b) What are the expected value and standard deviation of the difference (female minus male) between their scores?
(c) From the information given, can you find the probability that the woman chosen scores higher than the man? If so, find this probability. If not,
explain why you cannot.
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