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GEM2900, 2014/2015 Semester I Exercise Sheet 03

Exercise 1. An absent-minded secretary prepared five letters and envelopes to send to fivedi↵erent people. Then he randomly placed letters in the envelopes without checking. A matchoccurs if the letter and its envelope are addressed to the same person. What is the probabilityof the following events?

(i) At least one of the five letters and envelopes match.

(ii) Exactly four of the letters and envelopes match.

(iii) All five letters and envelopes match.

Exercise 2. Let E and F be two events with probabilities given as P (E) = 1/2 and P (F ) = 1.Determine if E and F are independent.

Exercise 3. A bag contains six balls all of di↵erent colors.

(i) Four balls are taken out of the bag, one at a time. What is the probability that the ballsselected will be cyan, magenta, yellow, and black in that order?

(ii) The four selected balls are returned to the bag and then a handful of four balls is takenout, all at once. What is the probability that the four balls are cyan, magenta, yellow,and black?

Exercise 4. An ectopic pregnancy is twice as likely to develop when the pregnant woman isa smoker as it is when she is a nonsmoker. If 32% of women of childbearing age are smokers,what percentage of women having ectopic pregnancies are smokers?

Exercise 5. Consider the following version of the game of craps:The player rolls two dice. If the sum on the first roll is 7 or 11, the player wins the gameimmediately. If the sum on the first roll is 2, 3, or 12, the player loses the game immediately.However, if the sum on the first roll is 4, 5, 6, 8, 9, or 10, then the two dice are rolled againand again until the sum is either 7 or 11 or the original value. If the original value is obtaineda second time before either 7 or 11 is obtained, then the player wins. If either 7 or 11 isobtained before the original value is obtained a second time, then the player loses. Determinethe probability that the player will win this game.

Exercise 6. Suppose Edward is a physician. One of his patients has a lump in her breast.Edward is almost certain that it is benign; in fact, he thinks there is only a 1% chance thatit is malignant. Just to be sure, he has the patient undergo a mammogram, which is a breastX-ray designed to detect cancer. From the medical literature it is known that mammogramsare 80% accurate for malignant lumps, and 90% accurate for the benign lumps.

Sadly, the mammogram for Edward’s patient is returned with the news that the lump is malig-nant. During the review appointment, Edward shared the result with his patient, and told thepatient “Since the mammogram is 80% accurate for malignant lumps, I’m sorry to say that itis very likely that you have breast cancer.”

The patient decided to seek a second opinion from another physician, Gary, in another facility,on her mammogram result. During the appointment Gary said, “Even though the result showedthat your lump is malignant, the chance for it to be truly malignant is low. In fact, there is amore than 90% chance that the lump is benign”.

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(i) Do you agree with Edward’s or Gary’s statement to the patient? Justify your answer witha probability calculation.

(ii) What is the “80% accurate” referring to in Edward’s statement to the patient?

(iii) Assume another patient developed a malignant lump 3 years ago, but was diagnosed tobe benign then. In fact, she has done annual mammogram screening to track the lumpfor the past 3 years. The results have been negative (benign) for the past 3 years. Shedid not realize the lump to be malignant until this year (year 4). Given the accuracy ofmammogram screening indicated in the medical literature, do you think this is very likelyto happen? Explain your answer.

Exercise 7. Consider the following statement pertaining to HIV testing: “In a high-risk pop-ulation, virtually all people tested positive will be truly infected. Among people in a low-riskgroup the false positives will outnumber the true positives. Thus, for every infected personcorrectly identified as belonging to the low risk group, an estimated 10 non-HIV carriers willtest positive”.

(i) Suppose Frank is in the low risk group but that he has just tested positive. What is thechance that this result is a ‘false positive’ according to the above statement?

(ii) Assume we are screening 5 people in the low risk population. The lab technician forgotto write the names of the patients on the blood samples after drawing their blood forthe HIV test. The technician now has 5 vials of blood sample and 5 name labels. Hehas decided to label each blood sample at random (of course, this should not happenin reality). What is the probability that at least one of the blood samples is correctlylabelled by the technician?

Exercise 8. The random walk theory of securities prices holds that price movements in non-overlapping intervals of time are independent of each other. Suppose that we record onlywhether the price is up or down each year, and that the probability that our portfolio risesin price in any one year is 0.65. (This probability is approximately correct for a portfoliocontaining equal dollar amounts of all common stocks listed on the New York Stock Exchange).

(i) What is the probability that the portfolio goes up for 3 consecutive years?

(ii) If you know that the portfolio has risen in price 2 years in a row, what probability do youassign to the event that it will go down next year?

(iii) What is the probability that the portfolio’s value moves in the same direction in both ofthe next 2 years?

Exercise 9. Consider the following variation of the Monty Hall game. There are 3 doorslabelled 1 to 3, with a car behind one door and goats behind the remaining 2 (each door isequally likely to hide the car). You are allowed to select a door. The host does not knowwhere the car is and chooses one of the remaining doors at random and opens it. If the hostreveals the car you lose immediately but if the host reveals a goat you are o↵ered the chance toswitch to the remaining unopened door. We will further assume that your aim is to win the carand that you know all of the above information (either because it is specified as rules or fromwatching previous games). For each of the strategies below, calculate your chance of winning:



(i) Initially you select door 1 and stick with it.

(ii) Initially you select door 1 and always switch when you are given the opportunity to switch.

(iii) Initially you select door 1 and if the host opens door 2 and you are given the opportunityto switch then you switch.

(iv) Suppose that initially you select door 1, and that the host opens door 2 to reveal a goat.Conditional on this information, what is your chance of winning if you switch?

Exercise 10. A manufacturer of cornflakes decides to include plastic figurines of famous statis-ticians in their cornflake packets in a marketing campaign to enhance the attractiveness of theirproduct to young people. Each packet of cornflakes contains one figurine, and there are 5 dif-ferent types of figurines to collect. The figurine in each packet is equally likely to be any of the5 types. A GEM2900 student hears about the promotion and immediately goes out to buy 6packets. What is the chance that she obtains all 5 figurines?

Exercise 11. In a certain town there are two bus companies. For company A all its buses areblue. For company B all its buses are red. Company A owns 30% of buses in the town andcompany B owns 70% of the buses. One night an accident occurs where a pedestrian is injuredand the bus drives away apparently without the driver being aware that any accident hasoccurred. A witness says that the bus involved was red. However, at the time of the accidentthe scene was poorly illuminated and a psychologist testifies that there is a 10% chance that ablue bus might be mistaken to be red, or that a red bus might be mistaken to be blue.

(i) What is the probability that the bus involved in the accident was red?

(ii) Suppose that the witness had identified the bus involved as being blue rather than red.What is the probability that the bus involved in the accident was blue?

(iii) If your answers to (i) and (ii) are di↵erent, explain why they are di↵erent. If they are thesame, explain why they are the same.

Exercise 12. n couples (husbands and wives) are invited to a party. We assume in what followsthat all the people invited have birthdays that are independent, that there are 365 days in ayear, and that any day is equally likely to be the birthday of a particular person.

(i) Write down an expression for the probability that at least two of the wives share the samebirthday.

(ii) Write down an expression for the probability that there exists at least one pair of coupleswhere both husbands share the same birthday and both wives share the same birthday(the day does not have to be the same for husbands and wives).

(iii) One of the n couples is the couple hosting the party. What is the chance that there existsone other couple in which the husband has the same birthday as the host husband andthe wife has the same birthday as the host wife?

(iv) For your answer in (iii), how large does n need to be to make the probability larger than0.5?

Exercise 13. A lecturer administers an exam to his class of 500 GEM2900 students. He dropsall the exams on the floor so that the answer booklets and the cover sheets identifying the



students are separated, so that he has no way of knowing which student submitted whichanswer booklet. He decides to pair up the answer books and cover sheets randomly. What isthe (approximate) probability that at least one student gets assigned their correct mark (thatis, what is the chance that at least one cover sheet and answer booklet are correctly matched)?

Answers

1. 0.6333; 0; 1/120

3. 0.0028; 0.067

4. 16/33

5. 0.448

7. 10/11; 0.6333

8. 0.275; 0.35; 0.545.

9. 1/3; 1/3; 1/3; 1/2

10. 0.1152

11. 0.95; 0.79

12. 94

13. 0.6321

Note: Solutions will be uploaded by the end of Week 8.


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