Question

17) According to insurance records a car with a certain protection system will be recovered 91% of the time. Find the probability that 5 of 6 stolen cars will be recovered. 18) The probability that a football game will go into overtime is 18%. What is the probability that two of three football games will go to into overtime? 19) Fifty percent of the people that use the Internet order something online. Find the probability that only two of 10 Internet users will order something online. 20) The probability that a house in an urban area will develop a leak is 6%. If 87 houses are randomly selected, what is the probability that none of the houses will develop a leak? 21) Sixty-five percent of men consider themselves knowledgeable soccer fans. If 13 men are randomly selected, find the probability that exactly five of them will consider themselves knowledgeable fans. 22) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of at most three girls in ten births. 23) Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Suppose that 950 couples each have a baby; find the mean and standard deviation for the number of boys in the 950 babies. 24) A quiz consists of 100 true or false questions. If the student guesses on each question, what is the standard deviation of the number of correct answers? 25) A quiz consists of 20 multiple choice questions, each with five possible answers, only one of which is correct. If a student guesses on each question, what is the mean and standard deviation of the number of correct answers?

Answer 1

17) The probability that 5 of 6 stolen cars will be recovered is given by the binomial probability formula:
`\( P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^(n - k) \)`, where n is the number of trials, k is the number of successes, and p is the probability of success. In this case, n = 6 , k = 5 , and p = 0.91 . Using the formula, the probability is approximately 0.315 .

18) The probability that two of three football games will go into overtime is also calculated using the binomial probability formula, where n = 3, k = 2 , and p = 0.18 . The probability is approximately 0.277 .

19) For the probability that only two of 10 Internet users will order something online, again use the binomial probability formula with n = 10 , k = 2 , and p = 0.5 . The probability is approximately 0.205 .

20) To find the probability that none of the 87 houses will develop a leak, use the binomial probability formula with n = 87 , k = 0 , and p = 0.94 (since the probability of not developing a leak is 1 - 0.06. The probability is approximately 0.058 .

21) Using the binomial probability formula with n = 13 , k = 5 , and p = 0.65, the probability that exactly five of the 13 men will consider themselves knowledgeable fans is approximately 0.205.

22) To find the probability of at most three girls in ten births, sum the probabilities of 0, 1, 2, and 3 girls using the binomial probability formula with n = 10 and p = 0.5 . The probability is approximately 0.999.

23) For the mean and standard deviation of the number of boys in 950 babies, the mean μ is given by μ = n. p and the standard deviation σ is given by σ = √n. p .(1 - p). Substituting n = 950 and p = 0.5 , we get μ = 475 and σ ≅ 15.44 .

24) The standard deviation of the number of correct answers in a quiz with 100 true or false questions is given by σ = √n . p . (1 - p). Substituting n = 100 and p = 0.5 , the standard deviation is σ = √25 = 5 .

25) The mean and standard deviation of the number of correct answers in a quiz with 20 multiple-choice questions can be found using the binomial distribution formula with n = 20 and p = 1/5 (since there is one correct answer out of five choices). The mean μ is n . p = 20 . 1/5 = 4, and the standard deviation σ is √n. p. (1 - p) = √20 . 1/5 . 4/5 = √16 = 4 .