91Ó°ÊÓ

Ms. Aquina has just had a biopsy on a possibly cancerous tumor. Not wanting to spoil a weekend family event, she does not want to hear any bad news in the next few days. But if she tells the doctor to call only if the news is good, then if the doctor does not call, Ms. Aquina can conclude that the news is bad. So, being a student of probability, Ms. Aquina instructs the doctor to flip a coin. If it comes up heads, the doctor is to call if the news is good and not call if the news is bad. If the coin comes up tails, the doctor is not to call. In this way, even if the doctor doesn't call, the news is not necessarily bad. Let \(\alpha\) be the probability that the tumor is cancerous; let \(\beta\) be the conditional probability that the tumor is cancerous given that the doctor does not call. (a) Which should be larger, \(\alpha\) or \(\beta ?\) (b) Find \(\beta\) in terms of \(\alpha,\) and prove your answer in part (a). 3.32. A family has \(j\) children with probability \(p_{j},\) where \(p_{1}=.1, p_{2}=.25, p_{3}=.35, p_{4}=.3 .\) A child from this fam- ily is randomly chosen. Given that this child is the eldest child in the family, find the conditional probability that the family has (a) only 1 child; (b) 4 children. Redo (a) and (b) when the randomly selected child is the youngest child of the family.

There are 3 coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time. When one of the 3 coins is selected at random and flipped, it shows heads. What is the probability that it was the two-headed coin?

Each of 2 cabinets identical in appearance has 2 drawers. Cabinet \(A\) contains a silver coin in each drawer, and cabinet \(B\) contains a silver coin in one of its drawers and a gold coin in the other. A cabinet is randomly selected, one of its drawers is opened, and a silver coin is found. What is the probability that there is a silver coin in the other drawer?

Suppose that an insurance company classifies people into one of three classes: good risks, average risks, and bad risks. The company's records indicate that the probabilities that good-, average-, and bad-risk persons will be involved in an accident over a 1 -year span are, respectively, .05, .15 and \(30 .\) If 20 percent of the population is a good risk, 50 percent an average risk, and 30 percent a bad risk, what proportion of people have accidents in a fixed year? If policyholder \(A\) had no accidents in \(2012,\) what is the probability that he or she is a good risk? is an average risk?

A simplified model for the movement of the price of a stock supposes that on each day the stock's price either moves up 1 unit with probability \(p\) or moves down 1 unit with probability \(1-p .\) The changes on different days are assumed to be independent. (a) What is the probability that after 2 days the stock will be at its original price? (b) What is the probability that after 3 days the stock's price will have increased by 1 unit? (c) Given that after 3 days the stock's price has increased by 1 unit, what is the probability that it went up on the first day?

Independent flips of a coin that lands on heads with probability \(p\) are made. What is the probability that the first four outcomes are (a) \(H, H, H, H ?\) (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H, H, H\) occurs before the pattern \(H, H, H, H ?\) Hint for part \((c):\) How can the pattern \(H, H, H, H\) occur first?

Genes relating to albinism are denoted by \(A\) and \(a\). Only those people who receive the \(a\) gene from both parents will be albino. Persons having the gene pair \(A, a\) are normal in appearance and, because they can pass on the trait to their offspring, are called carriers. Suppose that a normal couple has two children, exactly one of whom is an albino. Suppose that the nonalbino child mates with a person who is known to be a carrier for albinism. (a) What is the probability that their first offspring is an albino? (b) What is the conditional probability that their second offspring is an albino given that their firstborn is not?

Suppose that each child born to a couple is equally likely to be a boy or a girl, independently of the sex distribution of the other children in the family. For a couple having 5 children, compute the probabilities of the following events: (a) All children are of the same sex. (b) The 3 eldest are boys and the others girls. (c) Exactly 3 are boys. (d) The 2 oldest are girls. (e) There is at least 1 girl.

\(A\) and \(B\) alternate rolling a pair of dice, stopping either when \(A\) rolls the sum 9 or when \(B\) rolls the sum 6\. Assuming that \(A\) rolls first, find the probability that the final roll is made by \(A .\)

An investor owns shares in a stock whose present value is \(25 .\) She has decided that she must sell her stock if it goes either down to 10 or up to \(40 .\) If each change of price is either up 1 point with probability .55 or down 1 point with probability \(.45,\) and the successive changes are independent, what is the probability that the investor retires a winner?