# 4-3 Basic Skills and Concepts

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1 4-3 Basic Skills and Concepts Identifying Binomial Distributions. In Exercises 1 8, determine whether the given procedure results in a binomial distribution. For those that are not binomial, identify at least one requirement that is not satisfied. 1. Surveying people by asking them what they think of the current president 2. Surveying 1012 people and recording whether there is a should not response to this question: Do you think the cloning of humans should or should not be allowed? 3. Rolling a fair die 50 times 4. Rolling a loaded die 50 times and finding the number of times that 5 occurs 5. Recording the genders of 250 newborn babies 6. Determining whether each of 3000 heart pacemakers is acceptable or defective 7. Spinning a roulette wheel 12 times 8. Spinning a roulette wheel 12 times and finding the number of times that the outcome is an odd number 9. Finding Probabilities When Guessing Answers Multiple-choice questions each have five possible answers, one of which is correct. Assume that you guess the answers to three such questions.

3 188 C HAPTER 4 Probability Distributions 24. Find the probability that at least one American Airlines flight arrives on time. Is it unusual to not get at least one of six American Airlines flights arriving on time? 25. Color Blindness Nine percent of men and 0.25% of women cannot distinguish between the colors red and green. This is the type of color blindness that causes problems with traffic signals. If six men are randomly selected for a study of traffic signal perceptions, find the probability that exactly two of them cannot distinguish between red and green. 26. Acceptance Sampling The Telektronic Company purchases large shipments of fluorescent bulbs and uses this acceptance sampling plan: Randomly select and test 24 bulbs, then accept the whole batch if there is only one or none that doesn t work. If a particular shipment of thousands of bulbs actually has a 4% rate of defects, what is the probability that this whole shipment will be accepted? 27. IRS Audits The Hemingway Financial Company prepares tax returns for individuals. (Motto: We also write great fiction. ) According to the Internal Revenue Service, individuals making \$25,000 \$50,000 are audited at a rate of 1%. The Hemingway Company prepares five tax returns for individuals in that tax bracket, and three of them are audited. a. Find the probability that when five people making \$25,000 \$50,000 are randomly selected, exactly three of them are audited. b. Find the probability that at least three are audited. c. Based on the preceding results, what can you conclude about the Hemingway customers? Are they just unlucky, or are they being targeted for audits? 28. Directory Assistance An article in USA Today stated that Internal surveys paid for by directory assistance providers show that even the most accurate companies give out wrong numbers 15% of the time. Assume that you are testing such a provider by making 10 requests and also assume that the provider gives the wrong telephone number 15% of the time. a. Find the probability of getting one wrong number. b. Find the probability of getting at most one wrong number. c. If you do get at most one wrong number, does it appear that the rate of wrong numbers is not 15%, as claimed? 29. Overbooking Flights Air America has a policy of booking as many as 15 persons on an airplane that can seat only 14. (Past studies have revealed that only 85% of the booked passengers actually arrive for the flight.) Find the probability that if Air America books 15 persons, not enough seats will be available. Is this probability low enough so that overbooking is not a real concern for passengers? 30. Drug Reaction In a clinical test of the drug Viagra, it was found that 4% of those in a placebo group experienced headaches. a. Assuming that the same 4% rate applies to those taking Viagra, find the probability that among eight Viagra users, three experience headaches. b. Assuming that the same 4% rate applies to those taking Viagra, find the probability that among eight randomly selected users of Viagra, all eight experienced a headache. c. If all eight Viagra users were to experience a headache, would it appear that the headache rate for Viagra users is different than the 4% rate for those in the placebo group? Explain.

5 190 C HAPTER 4 Probability Distributions 4-3 Beyond the Basics 37. If a procedure meets all the conditions of a binomial distribution except that the number of trials is not fixed, then the geometric distribution can be used. The probability of getting the first success on the xth trial is given by P(x) 5 p(1 2 p) x21 where p is the probability of success on any one trial. Assume that the probability of a defective computer component is 0.2. Find the probability that the first defect is found in the seventh component tested. 38. If we sample from a small finite population without replacement, the binomial distribution should not be used because the events are not independent. If sampling is done without replacement and the outcomes belong to one of two types, we can use the hypergeometric distribution. If a population has A objects of one type, while the remaining B objects are of the other type, and if n objects are sampled without replacement, then the probability of getting x objects of type A and n 2 x objects of type B is Psxd 5 A! sa 2 xd!x!? B! sb 2 n 1 xd!sn 2 xd! 4 In Lotto 54, a bettor selects six numbers from 1 to 54 (without repetition), and a winning six-number combination is later randomly selected. Find the probability of getting a. All six winning numbers b. Exactly five of the winning numbers c. Exactly three of the winning numbers d. No winning numbers 39. The binomial distribution applies only to cases involving two types of outcomes, whereas the multinomial distribution involves more than two categories. Suppose we have three types of mutually exclusive outcomes denoted by A, B, and C. Let P(A) 5 p 1, P(B) 5 p 2, and P(C) 5 p 3. In n independent trials, the probability of x 1 outcomes of type A, x 2 outcomes of type B, and x 3 outcomes of type C is given by n! sx 1!dsx 2!dsx 3!d? p x 1 x 1? p 2 x 2? p 3 3 sa 1 Bd! sa 1 B 2 nd!n! A genetics experiment involves six mutually exclusive genotypes identified as A, B, C, D, E, and F, and they are all equally likely. If 20 offspring are tested, find the probability of getting exactly five A s, four B s, three C s, two D s, three E s, and three F s by expanding the above expression so that it applies to six types of outcomes instead of only three. 40. The Poisson distribution applies to occurrences of some event over a specified interval, such as time or distance. The probability of the event occurring x times over an interval is given by Psxd 5 mx? e 2m, where e x! and m is the mean number of occurrences over the interval. Over the past 100 years, the mean number of major earthquakes in the world for one year is Assuming that the Poisson distribution is a suitable model, find the probability that the number of earthquakes in a randomly selected year is a. 0 b. 1 c. 2 d. 3 e. 4 f. 5 g. 6 h. 7

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