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Antibiotic Resist ance. According to CDC estimates, at least \(2.8\) million people in the United States are sickened each year with antibiotic-resistant infections, and at least 35,000 die as a result. Antibiotic resistance occurs when disease-causing microbes become resistant to antibiotic drug therapy. Because this resistance is typically genetic and transferred to the next generations of microbes, it is a very serious public health problem. Of the infections considered most serious by the CDC, gonorrhea has an estimated \(1.14\) million new cases occurring annually, and approximately \(50 \%\) of those cases are resistant to any antibiotic. \({ }^{7}\) A public health clinic in California sees eight patients with gonorrhea in a given week. a. What is the distribution of \(X\), the number of these eight cases that are resistant to any antibiotic? b. What are the mean and standard deviation of \(X\) ? c. Find the probability that exactly one of the cases is resistant to any antibiotic. What is the probability that at least one case is resistant to any antibiotic? (Hint: It is easier to first find the probability that exactly zero of the eight cases were resistant.)

Roulette-Betting on Red. A roulette wheel has 38 slots, numbered 0,00 , and 1 to 36 . The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors. a. If you bet on "red," you win if the ball lands in a red slot. What is the probability of winning with a bet on red in a single play of roulette? b. You decide to play roulette four times, each time betting on red. What is the distribution of \(X\), the number of times you win? c. If you bet the same amount on each play and win on exactly two of the four plays, then you will "break even." What is the probability that you will break even? d. If you win on fewer than two of the four plays, then you will lose money. What is the probability that you will lose money?

False Positives in Testing for HIV. A rapid test for the presence in the blood of antibodies to HIV, the virus that causes AIDS, gives a positive result with probability about \(0.004\) when a person who is free of HIV antibodies is tested. A clinic test s 1000 people who are all free of HIV antibodies. a. What is the distribution of the number of positive tests? b. What is the mean number of positive tests? c. You cannot safely use the Normal approximation for this distribution. Explain why.