Chapter 5: Probability: What are the chances?

Drug testing Athletes are often tested for use of performance-enhancing drugs. Drug tests aren’t perfect—they sometimes say that an athlete took a

banned substance when that isn’t the case (a “false positive”). Other times, the test concludes that the athlete is “clean” when he or she actually took a

banned substance (a “false negative”). For one commonly used drug test, the probability of a false negative is 0.03.

(a) Interpret this probability as a long-run relative frequency.

(b) Which is a more serious error in this case: a false positive or a false negative? Justify your answer.

Preparing for the GMAT A company that offers courses to prepare students for the Graduate Management Admission Test (GMAT) has the following information about its customers: $20\%$ are currently undergraduate students in business; $15\%$ are undergraduate students in other fields of study; $60\%$ are

college graduates who are currently employed, and $5\%$are college graduates who are not employed. Choose a customer at random.

(a) What’s the probability that the customer is currently an undergraduate? Which rule of probability did you use to find the answer?

(b) What’s the probability that the customer is not an undergraduate business student? Which rule of probability did you use to find the answer?

Liar, liar! Sometimes police use a lie detector (also known as a polygraph) to help determine whether a suspect is telling the truth. A lie detector test isn’t foolproof—sometimes it suggests that a person is lying when they’re actually telling the truth (a “false positive”). Other times, the test says that the suspect is being truthful when the person is actually lying (a “false negative”). For one brand of polygraph machine, the probability of a false positive is $0.08$.
(a) Interpret this probability as a long-run relative frequency.
(b) Which is a more serious error in this case: a false positive or a false negative? Justify your answer.

Cold weather coming to A TV weather man, predicting a colder-than-normal winter, said, “First, in looking at the past few winters, there has been a lack of

really cold weather. Even though we are not supposed to use the law of averages, we are due.” Do you think that “due by the law of averages” makes sense in talking about the weather? Why or why not?

Testing the test Are false positives too common in some medical tests? Researchers conducted an experiment involving $250$ patients with a medical

condition and $750$ other patients who did not have the medical condition. The medical technicians who were reading the test results were unaware that they

were subjects in an experiment.

(a) Technicians correctly identified $240$ of the $250$patients with the condition. They also identified $50$ of the healthy patients as having the condition. What

were the false positive and false negative rates for the test?

(b) Given that a patient got a positive test result, what is the probability that the patient actually had the medical condition? Show your work.

The probability of a flush A poker player holds a flush when all $5$cards in the hand belong to the same suit. We will find the probability of a flush when $5$ cards are dealt. Remember that a deck contains $52$ cards, $13$ of each suit, and that when the deck is well shuffled, each card dealt is equally likely to be any of those that remain in the deck.

(a) We will concentrate on spades. What is the probability that the first card dealt is a spade? What is the conditional probability that the second card is

a spade given that the first is a spade?

(b) Continue to count the remaining cards to find the conditional probabilities of a spade on the third, the fourth, and the fifth card given in each case that

all previous cards are spades.

(c) The probability of being dealt 5 spades is the product of the five probabilities you have found. Why? What is this probability?

(d) The probability of being dealt $5$ hearts or $5$diamonds or $5$ clubs is the same as the probability of being dealt $5$ spades. What is the probability of being dealt a flush?

The geometric distributions You are tossing a pair of fair, six-sided dice in a board game. Tosses are independent. You land in a danger zone that

requires you to roll doubles (both faces showing the same number of spots) before you are allowed to play again. How long will you wait to play again?

(a) What is the probability of rolling doubles on a single toss of the dice? (If you need review, the possible outcomes appear in Figure 5.2 (page 300). All $36$ outcomes are equally likely.)

(b) What is the probability that you do not roll doubles on the first toss, but you do on the second toss?

(c) What is the probability that the first two tosses are not doubles and the third toss is doubles? This is the probability that the first doubles occurs on the

third toss.

(d) Now you see the pattern. What is the probability that the first doubles occurs on the fourth toss? On the fifth toss? Give the general result: what is the probability that the first doubles occurs on the kth toss?

A probability teaser Suppose (as is roughly correct) that each child born is equally likely to be a boy or a girl and that the genders of successive children are independent. If we let BG mean that the older child is a boy and the younger child is a girl, then each of the combinations BB, BG, GB, and GG has a probability $0.25$Ashley and Brianna each have two children.

(a) You know that at least one of Ashley’s children is a boy. What is the conditional probability that she has two boys?

(b) You know that Brianna’s older child is a boy. What is the conditional probability that she has two boys?

An athlete suspected of using steroids is given two tests that operate independently of each other. Test A has a probability $0.9$ of being positive if steroids have been used. Test B has a probability $0.8$ of being positive

if steroids have been used. What is the probability that neither test is positive if steroids have been used?

In an effort to find the source of an outbreak of food poisoning at a conference, a team of medical detectives carried out a study. They examined all $50$people who had food poisoning and a random sample of $200$people attending the conference who didn’t get food poisoning. The detectives found that $40\%$of the people with food poisoning went to a cocktail party on the second night of the conference, while only $10\%$of the people in the random sample attended the same party. Which of the following statements is appropriate for describing the $40\%$of people who went to the party? (Let F = got food poisoning and A = attended party.)

(a) P(F | A) = 0.40 (d) P(AC | F) = 0.40

(b) P(A | FC) = 0.40 (e) None of these

(c) P(F | AC) = 0.40