91Ó°ÊÓ

A chromosome mutation linked with colorblindness is known to occur, on the average, once in every ten thousand births. (a) Approximate the probability that exactly three of the next twenty thousand babies born will have the mutation. (b) How many babies out of the next twenty thousand would have to be born with the mutation to convince you that the "one in ten thousand" estimate is too low?

Assume that the number of hits, \(X\), that a baseball team makes in a nine- inning game has a Poisson distribution. If the probability that a team makes zero hits is \(\frac{1}{3}\), what are their chances of getting two or more hits?

Let \(X\) and \(Y\) be independent Poisson random variables with parameters \(\lambda\) and \(\mu\), respectively. Example \(3.12 .10\) established that \(X+Y\) is also Poisson with parameter \(\lambda+\mu\). Prove that same result using Theorem \(3.8 .3\).

Suppose that commercial airplane crashes in a certain country occur at the rate of \(2.5\) per year. (a) Is it reasonable to assume that such crashes are Poisson events? Explain. (b) What is the probability that four or more crashes will occur next year? (c) What is the probability that the next two crashes will occur within three months of one another?

Fifty spotlights have just been installed in an outdoor security system. According to the manufacturer's specifications, these particular lights are expected to burn out at the rate of \(1.1\) per one hundred hours. What is the expected number of bulbs that will fail to last for at least seventy-five hours?

State Tech’s basketball team, the Fighting Logarithms, have a 70% foul- shooting percentage. (a) Write a formula for the exact probability that out of their next one hundred free throws, they will make between seventy-five and eighty, inclusive. (b) Approximate the probability asked for in part (a)

Suppose that one hundred fair dice are tossed. Estimate the probability that the sum of the faces showing exceeds 370 . Include a continuity correction in your analysis.

Let \(X\) be the amount won or lost in betting \(\$ 5\) on red in roulette. Then \(p_{x}(5)=\frac{18}{38}\) and \(p_{x}(-5)=\frac{20}{38}\). If a gambler bets on red one hundred times, use the Central Limit Theorem to estimate the probability that those wagers result in less than \(\$ 50\) in losses.

Suppose \(X_{1}, X_{2}, X_{3}\), and \(X_{4}\) are independent Poisson random variables, each with parameter \(\lambda=3\). Let \(S=X_{1}+X_{2}+X_{3}+X_{4}\) (a) Use the Central Limit Theorem to approximate the probability that \(13 \leq S \leq 14\). (b) Calculate the exact probability that \(13 \leq S \leq 14\).

Among the many letters sent to a popular adviceto-the-lovelorn columnist, was one involving a paternity issue that raised an interesting statistical question. The distraught writer-call her "San Diego Reader" - said her husband is in the military and that she got pregnant the last day before he left for an extended tour of duty. Ten months and four days later the baby was born - usually a happy occasion - but her husband, accustomed to pregnancies being nine months long, became obsessed with the possibility that he might not be the child's father. DNA testing was not yet available. The only relevant information known at the time was that pregnancy durations are normally distributed with a mean ( \(\mu\) ) of 266 days and a standard deviation \((\sigma)\) of 16 days. For the benefit of San Diego Reader's husband, how would you associate a probability with a pregnancy lasting 10 months and 4 days? Do you think San Diego Reader is telling the truth?