Probability

Abstract

The term probability refers to the study of randomness and uncertainty. In any situation in which one of a number of possible outcomes may occur, the theory of probability provides methods for quantifying the chances, or likelihoods, associated with the various outcomes. The language of probability is constantly used in an informal manner in both written and spoken contexts. Examples include such statements as “It is likely that the Dow Jones Industrial Average will increase by the end of the year,” “There is a 50–50 chance that the incumbent will seek reelection,” “There will probably be at least one section of that course offered next year,” “The odds favor a quick settlement of the strike,” and “It is expected that at least 20,000 concert tickets will be sold.” In this chapter, we introduce some elementary probability concepts, indicate how probabilities can be interpreted, and show how the rules of probability can be applied to compute the probabilities of many interesting events. The methodology of probability will then permit us to express in precise language such informal statements as those given above.

Supplementary Exercises: (113–140)

• 113. The undergraduate statistics club at a certain university has 24 members.

  • a. All 24 members of the club are eligible to attend a conference next week, but they can only afford to send 4 people. In how many possible ways could 4 attendees be selected?

  • b. All club members are eligible for any of the four positions of president, VP, secretary, or treasurer. In how many possible ways can these positions be occupied?

  • c. Suppose it’s agreed that two people will be cochairs, one person secretary, and one person treasurer. How many ways are there to fill these positions now?

• 114. A small manufacturing company will start operating a night shift. There are 20 machinists employed by the company.

  • a. If a night crew consists of 3 machinists, how many different crews are possible?

  • b. If the machinists are ranked 1, 2, …, 20 in order of competence, how many of these crews would not have the best machinist?

  • c. How many of the crews would have at least 1 of the 10 best machinists?

  • d. If a 3-person crew is selected at random to work on a particular night, what is the probability that the best machinist will not work that night?

• 115. A factory uses three production lines to manufacture cans of a certain type. The accompanying table gives percentages of nonconforming cans, categorized by type of nonconformance, for each of the three lines during a particular time period.

  	Line 1	Line 2	Line 3
  Blemish	15	12	20
  Crack	50	44	40
  Pull Tab Problem	21	28	24
  Surface Defect	10	8	15
  Other	4	8	2

  During this period, line 1 produced 500 nonconforming cans, line 2 produced 400 such cans, and line 3 was responsible for 600 nonconforming cans. Suppose that one of these 1500 cans is randomly selected.

  • a. What is the probability that the can was produced by line 1? That the reason for nonconformance is a crack?

  • b. If the selected can come from line 1, what is the probability that it had a blemish?

  • c. Given that the selected can had a surface defect, what is the probability that it came from line 1?

• 116. An employee of the records office at a university currently has ten forms on his desk awaiting processing. Six of these are withdrawal petitions, and the other four are course substitution requests.

  • a. If he randomly selects six of these forms to give to a subordinate, what is the probability that only one of the two types of forms remains on his desk?

  • b. Suppose he has time to process only four of these forms before leaving for the day. If these four are randomly selected one by one, what is the probability that each succeeding form is of a different type from its predecessor?

• 117. One satellite is scheduled to be launched from Cape Canaveral in Florida, and another launching is scheduled for Vandenberg Air Force Base in California. Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral launch goes off on schedule. If A and B are independent events with P(A) > P(B) and P(A∪B) = .626, P(A∩B) = .144, determine the values of P(A) and P(B).

• 118. A transmitter is sending a message by using a binary code, namely a sequence of 0’s and 1’s. Each transmitted bit (0 or 1) must pass through three relays to reach the receiver. At each relay, the probability is .2 that the bit sent will be different from the bit received (a reversal). Assume that the relays operate independently of one another.

  $$ {\text{Transmitter}} \to {\text{Relay 1}} \to {\text{Relay 2}} \to {\text{Relay 3 }} \to {\text{ Receiver}} $$

  • a. If a 1 is sent from the transmitter, what is the probability that a 1 is sent by all three relays?

  • b. If a 1 is sent from the transmitter, what is the probability that a 1 is received by the receiver? [Hint: The eight experimental outcomes can be displayed on a tree diagram with three generations of branches, one generation for each relay.]

  • c. Suppose 70% of all bits sent from the transmitter are 1’s. If a 1 is received by the receiver, what is the probability that a 1 was sent?

• 119. Individual A has a circle of five close friends (B, C, D, E, and F). A has heard a certain rumor from outside the circle and has invited the five friends to a party to circulate the rumor. To begin, A selects one of the five at random and tells the rumor to the chosen individual. That individual then selects at random one of the four remaining individuals and repeats the rumor. Continuing, a new individual is selected from those not already having heard the rumor by the individual who has just heard it, until everyone has been told.

  • a. What is the probability that the rumor is repeated in the order B, C, D, E, and F?

  • b. What is the probability that F is the third person at the party to be told the rumor?

  • c. What is the probability that F is the last person to hear the rumor?

• 120. Refer to the previous exercise. If at each stage the person who currently “has” the rumor does not know who has already heard it and selects the next recipient at random from all five possible individuals, what is the probability that F has still not heard the rumor after it has been told ten times at the party?

• 121. A chemist is interested in determining whether a certain trace impurity is present in a product. An experiment has a probability of .80 of detecting the impurity if it is present. The probability of not detecting the impurity if it is absent is .90. The prior probabilities of the impurity being present and being absent are .40 and .60, respectively. Three separate experiments result in only two detections. What is the posterior probability that the impurity is present?

• 122. Fasteners used in aircraft manufacturing are slightly crimped so that they lock enough to avoid loosening during vibration. Suppose that 95% of all fasteners pass an initial inspection. Of the 5% that fail, 20% are so seriously defective that they must be scrapped. The remaining fasteners are sent to a re-crimping operation, where 40% cannot be salvaged and are discarded. The other 60% of these fasteners are corrected by the re-crimping process and subsequently pass inspection.

  • a. What is the probability that a randomly selected incoming fastener will pass inspection either initially or after re-crimping?

  • b. Given that a fastener passed inspection, what is the probability that it passed the initial inspection and did not need re-crimping?

• 123. One percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 90% detection rate for carriers and a 5% detection rate for noncarriers. Suppose the test is applied independently to two different blood samples from the same randomly selected individual.

  • a. What is the probability that both tests yield the same result?

  • b. If both tests are positive, what is the probability that the selected individual is a carrier?

• 124. A system consists of two components. The probability that the second component functions in a satisfactory manner during its design life is .9, the probability that at least one of the two components does so is .96, and the probability that both components do so is .75. Given that the first component functions in a satisfactory manner throughout its design life, what is the probability that the second one does also?

• 125. A certain company sends 40% of its overnight mail parcels via express mail service E₁. Of these parcels, 2% arrive after the guaranteed delivery time (denote the event “late delivery” by L). If a record of an overnight mailing is randomly selected from the company’s file, what is the probability that the parcel went via E₁ and was late?

• 126. Refer to the previous exercise. Suppose that 50% of the overnight parcels are sent via express mail service E₂ and the remaining 10% are sent via E₃. Of those sent via E₂, only 1% arrive late, whereas 5% of the parcels handled by E₃ arrive late.

  • a. What is the probability that a randomly selected parcel arrived late?

  • b. If a randomly selected parcel has arrived on time, what is the probability that it was not sent via E₁?

• 127. A company uses three different assembly lines—A₁, A₂, and A₃—to manufacture a particular component. Of those manufactured by line A₁, 5% need rework to remedy a defect, whereas 8% of A₂’s components need rework and 10% of A₃’s need rework. Suppose that 50% of all components are produced by line A₁, 30% are produced by line A₂, and 20% come from line A₃. If a randomly selected component needs rework, what is the probability that it came from line A₁? From line A₂? From line A₃?

• 128. Disregarding the possibility of a February 29 birthday, suppose a randomly selected individual is equally likely to have been born on any one of the other 365 days. If ten people are randomly selected, what is the probability that either at least two have the same birthday or at least two have the same last three digits of their Social Security numbers? [Note: The article “Methods for Studying Coincidences” (F. Mosteller and P. Diaconis, J. Amer. Statist. Assoc. 1989: 853–861) discusses problems of this type.]

• 129. One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the sun’s energy reflected from the rock surface. Let R₁, R₂, and R₃ denote measured spectrum intensities at three different wavelengths; typically, for granite R₁ < R₂ < R₃, whereas for basalt R₃ < R₁ < R₂. When measurements are made remotely (using aircraft), various orderings of the R_i’s may arise whether the rock is basalt or granite. Flights over regions of known composition have yielded the following information:

  	Granite	Basalt
  R1 < R2 < R3	60%	10%
  R1 < R3 < R2	25%	20%
  R3 < R1 < R2	15%	70%

  Suppose that for a randomly selected rock specimen in a certain region, P(granite) = .25 and P(basalt) = .75.

  • a. Show that P(granite | R₁ < R₂ < R₃) > P(basalt | R₁ < R₂ < R₃). If measurements yielded R₁ < R₂ < R₃, would you classify the rock as granite or basalt?

  • b. If measurements yielded R₁ < R₃ < R₂, how would you classify the rock? Answer the same question for R₃ < R₁ < R₂.

  • c. Using the classification rules indicated in parts (a) and (b), when selecting a rock from this region, what is the probability of an erroneous classification? [Hint: Either G could be classified as B or B as G, and P(B) and P(G) are known.]

  • d. If P(granite) = p rather than .25, are there values of p (other than 1) for which a rock would always be classified as granite?

• 130. In a Little League baseball game, team A’s pitcher throws a strike 50% of the time and a ball 50% of the time, successive pitches are independent of each other, and the pitcher never hits a batter. Knowing this, team B’s manager has instructed the first batter not to swing at anything. Calculate the probability that

  • a. The batter walks on the fourth pitch.

  • b. The batter walks on the sixth pitch (so two of the first five must be strikes), using a counting argument or constructing a tree diagram.

  • c. The batter walks.

  • d. The first batter up scores while no one is out (assuming that each batter pursues a no-swing strategy).

• 131. The Matching Problem. Four friends—Allison, Beth, Carol, and Diane—who have identical calculators are studying for a statistics exam. They set their calculators down in a pile before taking a study break and then pick them up in random order when they return from the break.

  • a. What is the probability all four friends pick up the correct calculator?

  • b. What is the probability that at least one of the four gets her own calculator? [Hint: Let A be the event that Alice gets her own calculator, and define events B, C, and D analogously for the other three students. How can the event {at least one gets her own calculator} be expressed in terms of the four events A, B, C, and D? Now use a general law of probability.]

  • c. Generalize the answer from part (b) to n individuals. Can you recognize the result when n is large (the approximation to the resulting series)?

• 132. A particular airline has 10 a.m. flights from Chicago to New York, Atlanta, and Los Angeles. Let A denote the event that the New York flight is full and define events B and C analogously for the other two flights. Suppose P(A) = .6, P(B) = .5, P(C) = .4 and the three events are independent. What is the probability that

  • a. All three flights are full? That at least one flight is not full?

  • b. Only the New York flight is full? That exactly one of the three flights is full?

• 133. The Secretary Problem. A personnel manager is to interview four candidates for a job. These are ranked 1, 2, 3, and 4 in order of preference and will be interviewed in random order. However, at the conclusion of each interview, the manager will know only how the current candidate compares to those previously interviewed. For example, the interview order 3, 4, 1, 2 generates no information after the first interview and shows that the second candidate is worse than the first, and that the third is better than the first two. However, the order 3, 4, 2, 1 would generate the same information after each of the first three interviews. The manager wants to hire the best candidate but must make an irrevocable hire/no hire decision after each interview. Consider the following strategy: Automatically reject the first s candidates, and then hire the first subsequent candidate who is best among those already interviewed (if no such candidate appears, the last one interviewed is hired).

  For example, with s = 2, the order 3, 4, 1, 2 would result in the best being hired, whereas the order 3, 1, 2, 4 would not. Of the four possible s values (0, 1, 2, and 3), which one maximizes P(best is hired)? [Hint: Write out the 24 equally likely interview orderings; s = 0 means that the first candidate is automatically hired.]

• 134. Consider four independent events A₁, A₂, A₃, and A₄ and let p_i  = P(A_i) for i = 1, 2, 3, 4. Express the probability that at least one of these four events occurs in terms of the p_i’s, and do the same for the probability that at least two of the events occur.

• 135. A box contains the following four slips of paper, each having exactly the same dimensions: (1) win prize 1; (2) win prize 2; (3) win prize 3; (4) win prizes 1, 2, and 3. One slip will be randomly selected. Let A₁ = {win prize 1}, A₂ = {win prize 2}, and A₃ = {win prize 3}. Show that A₁ and A₂ are independent, that A₁ and A₃ are independent, and that A₂ and A₃ are also independent (this is pairwise independence). However, show that P(A₁∩A₂∩A₃) ≠ P(A₁) · P(A₂) · P(A₃), so the three events are not mutually independent.

• 136. Consider a woman whose brother is afflicted with hemophilia, which implies that the woman’s mother has the hemophilia gene on one of her two X chromosomes (almost surely not both, since that is generally fatal). Thus there is a 50–50 chance that the woman’s mother has passed on the bad gene to her. The woman has two sons, each of whom will independently inherit the gene from one of her two chromosomes. If the woman herself has a bad gene, there is a 50–50 chance she will pass this on to a son. Suppose that neither of her two sons is afflicted with hemophilia. What then is the probability that the woman is indeed the carrier of the hemophilia gene? What is this probability if she has a third son who is also not afflicted?

• 137. Jurors may be a priori biased for or against the prosecution in a criminal trial. Each juror is questioned by both the prosecution and the defense (the voir dire process), but this may not reveal bias. Even if bias is revealed, the judge may not excuse the juror for cause because of the narrow legal definition of bias. For a randomly selected candidate for the jury, define events B₀, B₁, and B₂ as the juror being unbiased, biased against the prosecution, and biased against the defense, respectively. Also let C be the event that bias is revealed during the questioning and D be the event that the juror is eliminated for cause. Let b_i  = P(B_i) (i = 0, 1, 2), c = P(C|B₁) = P(C|B₂), and d = P(D | B₁∩C) = P(D | B₂∩C) [“Fair Number of Peremptory Challenges in Jury Trials,” J. Amer. Statist. Assoc. 1979: 747–753].

  • a. If a juror survives the voir dire process, what is the probability that he/she is unbiased (in terms of the b_i’s, c, and d)? What is the probability that he/she is biased against the prosecution? What is the probability that he/she is biased against the defense? [Hint: Represent this situation using a tree diagram with three generations of branches.]

  • b. What are the probabilities requested in (a) if b₀ = .50, b₁ = .10, b₂ = .40 (all based on data relating to the famous trial of the Florida murderer Ted Bundy), c = .85 (corresponding to the extensive questioning appropriate in a capital case), and d = .7 (a “moderate” judge)?

• 138. Gambler’s Ruin. Allan and Beth currently have $2 and $3, respectively. A fair coin is tossed. If the result of the toss is H, Allan wins $1 from Beth, whereas if the coin toss results in T, then Beth wins $1 from Allan. This process is then repeated, with a coin toss followed by the exchange of $1, until one of the two players goes broke (one of the two gamblers is ruined). We wish to determine

  $$ a_{2} \, = \,P\left( {{\text{Allan is the winner}}|{\text{he starts with}}\;\$ 2} \right) $$

  To do so, let’s also consider probabilities

  $$ a_{i} \, = \,P\left( {{\text{Allan wins}}|{\rm{he\, starts\, with}}\;\$ i} \right)\ \text{for} \; i = 0, 1, 3, 4, \,\rm{and}\, 5.$$

  • a. What are the values of a₀ and a₅?

  • b. Use the Law of Total Probability to obtain an equation relating a₂ to a₁ and a₃. [Hint: Condition on the result of the first coin toss, realizing that if it is a H, then from that point Allan starts with $3.]

  • c. Using the logic described in (b), develop a system of equations relating a_i (i = 1, 2, 3, 4) to a_i–1 and a_i+1. Then solve these equations. [Hint: Write each equation so that a_i − a_i–1 is on the left-hand side. Then use the result of the first equation to express each other a_i − a_i–1 as a function of a₁, and add together all four of these expressions (i = 2, 3, 4, 5).]

  • d. Generalize the result to the situation in which Allan’s initial fortune is $a and Beth’s is $b. [Note: The solution is a bit more complicated if p = P(Allan wins $1) ≠ .5.]

• 139. An event A is said to attract event B if P(B | A) > P(B) and repel B if P(B | A) < P(B). (This refines the notion of dependent events by specifying whether A makes B more likely or less likely to occur.)

  • a. Show that if A attracts B, then A repels B′.

  • b. Show that if A attracts B, then A′ repels B.

  • c. Prove the Law of Mutual Attraction: event A attracts event B if, and only if, B attracts A.

• 140. A fair coin is tossed repeatedly until either the sequence TTH or the sequence THT is observed. Let B be the event that stopping occurs because TTH was observed (i.e., that TTH is observed before THT). Calculate P(B). [Hint: Consider the following partition of the sample space: A₁ = {1st toss is H}, A₂ = {1st two tosses are TT}, A₃ = {1st three tosses are THT}, and A₄ = {1st three tosses are THH}. Also denote P(B) by p. Apply the Law of Total Probability, and p will appear on both sides in various places. The resulting equation is easily solved for p.]