91Ó°ÊÓ

Every day Jo practices her tennis serve by continually serving until she has had a total of 50 successful serves. If each of her serves is, independently of previous ones, successful with probability \(.4,\) approximately what is the probability that she will need more than 100 serves to accomplish her goal? Hint: Imagine even if Jo is successful that she continues to serve until she has served exactly 100 times. What must be true about her first 100 serves if she is to reach her goal?

One thousand independent rolls of a fair die will be made. Compute an approximation to the probability that the number 6 will appear between 150 and 200 times inclusively. If the number 6 appears exactly 200 times, find the probability that the number 5 will appear less than 150 times.

The lifetimes of interactive computer chips produced by a certain semiconductor manufacturer are normally distributed with parameters \(\mu=1.4 \times 10^{6}\) hours and \(\sigma=\) \(3 \times 10^{5}\) hours. What is the approximate probability that a batch of 100 chips will contain at least 20 whose lifetimes are less than \(1.8 \times 10^{6} ?\)

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability .95 Approximate the probability that at most 10 of the next 150 items produced are unacceptable.

In 10,000 independent tosses of a coin, the coin landed on heads 5800 times. Is it reasonable to assume that the coin is not fair? Explain.

A model for the movement of a stock supposes that if the present price of the stock is \(s,\) then after one period, it will be either \(u s\) with probability \(p\) or \(d s\) with probability \(1-p .\) Assuming that successive movements are independent, approximate the probability that the stock's price will be up at least 30 percent after the next 1000 periods if \(u=1.012, d=0.990,\) and \(p=.52\).

The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter \(\lambda=\frac{1}{2} .\) What is (a) the probability that a repair time exceeds 2 hours? (b) the conditional probability that a repair takes at least 10 hours, given that its duration exceeds 9 hours?

The number of years a radio functions is exponentially distributed with parameter \(\lambda=\frac{1}{8} .\) If Jones buys a used radio, what is the probability that it will be working after an additional 8 years?

The lung cancer hazard rate \(\lambda(t)\) of a \(t\) -year-old male smoker is such that $$ \lambda(t)=.027+.00025(t-40)^{2} \quad t \geq 40 $$ Assuming that a 40 -year-old male smoker survives all other hazards, what is the probability that he survives to (a) age 50 and (b) age 60 without contracting lung cancer?

Let \(Y\) be a lognormal random variable (see Example 7e for its definition) and let \(c>0\) be a constant. Answer true or false to the following, and then give an explanation for your answer. (a) \(c Y\) is lognormal; (b) \(c+Y\) is lognormal.