D ESCRIPTION OF D ATA
3.5 Some Elementary Theorems
With the use of mathematical induction, the third axiom of probability can be extended to include any number of mutually exclusive events; in other words, the following can be shown.
Theorem 3.4 If A1, A2, . . . , Anare mutually exclusive events in a sample space S, then
P( A1∪ A2∪ · · · ∪ An)= P( A1)+ P( A2)+ · · · + P( An) Generalization of the third
axiom of probability
In the next chapter we shall see how the third axiom of probability must be modified so that the axioms apply also to sample spaces which are not finite.
EXAMPLE 15 Probabilities add for mutually exclusive events
The probability that a consumer testing service will rate a new antipollution device for cars very poor, poor, fair, good, very good, or excellent are 0.07, 0.12, 0.17, 0.32, 0.21, and 0.11. What are the probabilities that it will rate the device
(a) very poor, poor, fair, or good;
(b) good, very good, or excellent?
Solution Since the probabilities are all mutually exclusive, direct substitution into the formula of Theorem 3.4 yields
0.07 + 0.12 + 0.17 + 0.32 = 0.68 for part (a) and
0.32 + 0.21 + 0.11 = 0.64
for part (b). j
As it can be shown that a sample space of n points (outcomes) has 2nsubsets, it would seem that the problem of specifying a probability function (namely, a proba-bility for each subset or event) can easily become very tedious. Indeed, for n= 20 there are already more than 1 million possible events. Fortunately, this task can be simplified considerably by the use of the following theorem:
Theorem 3.5 If A is an event in the finite sample spaceS, then P(A) equals the sum of the probabilities of the individual outcomes comprising A.
Rule for calculating probability of an event
To prove this theorem, let E1, E2, . . . , Enbe the n outcomes comprising event A, so that we can write A = E1 ∪ E2 ∪ · · · ∪ En. Since the E’s are individual outcomes, they are mutually exclusive, and by Theorem 3.4 we have
P(A)= P(E1∪ E2∪ · · · ∪ En)
= P(E1)+ P(E2)+ · · · + P(En) which completes the proof.
EXAMPLE 16 Using a Venn diagram to visualize probability calculations
Refer to the used car classification example on page 60. Suppose that the proba-bilities of the 18 outcomes are as shown in Figure 3.9 (which, except for the the probabilities, is identical to Figure 3.5).
Sec 3.5 Some Elementary Theorems 73
Solution Adding the probabilities of the outcomes comprising the respective events, we get P(M1)= 0.03 + 0.06 + 0.07 + 0.02 + 0.01 + 0.01 = 0.20
In Theorem 3.4 we saw that the third axiom of probability can be extended to include more than two mutually exclusive events. Another useful and important extension of this axiom allows us to find the probability of the union of any two events inS regardless of whether or not they are mutually exclusive. To motivate the theorem which follows, let us consider the Venn diagram of Figure 3.10, which concerns the job offers received by recent engineering-school graduates. The letters I and G stand for a job offer from industry and a job offer from the government, respectively.
It follows from the Venn diagram that
P(I)= 0.18 + 0.12 = 0.30 P(G)= 0.12 + 0.24 = 0.36
Figure 3.10
Venn diagram for job offers
0.18 0.12 0.24
I G
S
and
P(I∪ G) = 0.18 + 0.12 + 0.24 = 0.54
We were able to add the various probabilities because they represent mutually ex-clusive events.
Had we erroneously used the third axiom of probability to calculate P(I∪ G), we would have obtained P(I)+ P(G) = 0.30 + 0.36, which exceeds the correct value by 0.12. This error results from adding in P(I∩ G) twice, once in P(I) = 0.30 and once in P(G)= 0.36 and, we could correct for it by subtracting 0.12 from 0.66.
Thus, we would get
P(I∪ G) = P(I) + P(G) − P(I ∩ G)
= 0.30 + 0.36 − 0.12
= 0.54
and this agrees, as it should, with the result obtained before.
In line with this motivation, let us now state and prove the following theorem:
Theorem 3.6 If A and B are any events inS, then P(A∪ B) = P(A) + P(B) − P(A ∩ B) General addition rule for
probability
To prove this theorem,
P(A∪ B) = P(A ∩ B) + P(A ∩ B ) + P( A ∩ B)
= [P(A ∩ B) + P(A ∩ B )]
+ [P(A ∩ B) + P( A ∩ B)] − P(A ∩ B)
= P(A) + P(B) − P(A ∩ B).
where, in the third line, we add and subtract P(A∩ B). Note that when A and B are mutually exclusive so that P(A∩ B) = 0, Theorem 3.6 reduces to the third axiom of probability. For this reason, we sometimes refer to the third axiom of probability as the special addition rule.
EXAMPLE 17 Using the general addition rule for probability
With reference to the used car example of page 60, find the probability that a car will have low mileage or be expensive to operate, namely P(M1∪ C3).
Solution Making use of the results obtained on page 73, P(M1) = 0.20, P(C3) = 0.40, and P(M1 ∩ C3) = 0.08, we substitute into the general addition rule of
Sec 3.5 Some Elementary Theorems 75
Theorem 3.6 to get
P(M1∪ C3)= P(M1)+ P(C3)− P(M1∩ C3)
= 0.20 + 0.40 − 0.08
= 0.52 j
EXAMPLE 18 The probability of requiring repair under warranty
If the probabilities are 0.87, 0.36, and 0.29 that, while under warranty, a new car will require repairs on the engine, drive train, or both, what is the probability that a car will require one or the other or both kinds of repairs under the warranty?
Solution Substituting these given values into the formula of Theorem 3.6, we get
0.87 + 0.36 − 0.29 = 0.94 j
Note that the general addition rule, Theorem 3.6, can be generalized further so that it applies to more than two events (see Exercise 3.49).
Using axioms of probability, we can derive many other theorems which play important roles in applications. For instance, let us show the following:
Theorem 3.7 If A is any event inS, then P( A ) = 1 − P(A).
Probability rule of the complement
To prove this theorem, we make use of the fact that A and A are mutually exclusive by definition, and that A∪ A = S (namely, that among them A and A contain all the elements inS). Hence we can write
P(A)+ P( A ) = P(A ∪ A )
= P(S)
= 1
so that P( A )= 1−P(A). As a special case we find that P(φ) = 1−P(S) = 0 since the empty setφ is the complement of S.
EXAMPLE 19 Using the probability rule of the complement
Referring to the used car example of page 60 and the results on page 73, find (a) the probability that a used car will not have low mileage
(b) the probability that a used car will either not have low mileage or not be expensive to operate
Solution By the rule of the complement
(a) P( M1)= 1 − P(M1)= 1 − 0.20 = 0.80
(b) Since M1∪ C3= M1∩ C3by the rule of the complement we get
P( M1∪ C3)= 1 − P(M1∩ C3)= 1 − 0.08 = 0.92 j
Exercises
3.28 (a) Among 880 smart phones sold by a retailer, 72 required repairs under the warranty. Estimate the probability that a new phone, which has just been sold, will require repairs under the warranty. Ex-plain your reasoning.
(b) Last year 8,400 students applied for the 6,000 student season tickets available for football games. Next year you will apply and would like to estimate the probability of receiv-ing a season ticket. Give your estimate and
comment on one factor that might influence the accuracy of your estimate.
3.29 When we roll a pair of balanced dice, what are the probabilities of getting
3.30 The registration numbers for the candidates of an en-trance test are numbered from 000001 to 200000.
What is the probability that a candidate will get a reg-istration number divisible by 40?
3.31 A car rental agency has 19 compact cars and 12 intermediate-size cars. If four of the cars are randomly selected for a safety check, what is the probability of getting two of each kind?
3.32 Last year; the maximum daily temperature in a plants’
server room exceeded 68◦F in 12 days. Estimate the probability that the maximum temperature will exceed 68◦F tomorrow.
3.33 In a group of 160 graduate engineering students, 92 are enrolled in an advanced course in statistics, 63 are enrolled in a course in operations research, and 40 are enrolled in both. How many of these students are not enrolled in either course?
3.34 Among 150 persons interviewed as part of an urban mass transportation study, some live more than 3 miles from the center of the city (A), some now regularly drive their own car to work (B), and some would gladly switch to public mass transportation if it were available (C). Use the information given in Figure 3.11 to find (a) N(A); Figure 3.11 Diagram for Exercise 3.34
(e) N(A∩ C);
3.35 An experiment has the four possible mutually exclu-sive outcomes A, B,C, and D. Check whether the fol-lowing assignments of probability are permissible:
(a) P(A)= 0.38, P(B) = 0.16, P(C) = 0.11, P(D) = 3.36 With reference to Exercise 3.1, suppose that the points
(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 1), (2, 2), (2, 3), (3, 0), (3, 1), (3, 2), and (3, 3) have the probabilities 0.080, 0.032, 0.086, 0.064, 0.085, 0.073, 0.065, 0.091, 0.071, 0.050, 0.046, 0.075, 0.040, 0.021, 0.080, and 0.041.
(a) Verify that this assignment of probabilities is permissible.
(b) Find the probabilities of events A, B, and C given in part (b) of that exercise.
(c) Calculate the probabilities that one, two, or three supplies of white cement are adulterated.
3.37 With reference to Exercise 3.7, suppose that each point (i, j) of the sample space is assigned the probability 420/401
2(i+ j).
(a) Verify that this assignment of probabilities is permissible.
(b) Find the probabilities of events X , Y , and Z described in part (b) of that exercise.
(c) Find the probabilities that two, three, or four of the supervisors will be present on the site.
3.38 Explain why there must be a mistake in each of the following statements:
(a) The probability that a mineral sample will contain silver is 0.38 and the probability that it will not contain silver is 0.52.
(b) The probability that a drilling operation will be a success is 0.34 and the probability that it will not be a success is−0.66.
Sec 3.5 Some Elementary Theorems 77
(c) An air-conditioning repair person claims that the probability is 0.82 that the compressor is all right, 0.64 that the fan motor is all right, and 0.41 that they are both all right.
3.39 Refer to parts (d) and (c) of Exercise 3.13 to show that (a) P(A∩ B) ≤ P(A);
(b) P(A∪ B) ≥ P(A).
3.40 Explain why there must be a mistake in each of the following statements:
(a) The probability that a student will get an A in a geology course is 0.3, and the probability that he or she will get either an A or a B is 0.27.
(b) A company is working on the construction of two shopping centers; the probability that the larger one will be completed on time is 0.35 and the probability that both will be completed on time is 0.42.
3.41 If A and B are mutually exclusive events, P(A)= 0.45, and P(B)= 0.30, find
(a) P( A );
(b) P( A∪ B );
(c) P( A∩ B );
(d) P( A∩ B ).
3.42 With reference to Exercise 3.34, suppose that the ques-tionnaire filled in by one of the 150 persons is to be double-checked. If it is chosen in such a way that each questionnaire has a probability of 1
150 of being se-lected, find the probabilities that the person
(a) lives more than 3 miles from the center of the city;
(b) regularly drives his or her car to work;
(c) does not live more than 3 miles from the center of the city and would not want to switch to public mass transportation if it were available;
(d) regularly drives his or her car to work but would gladly switch to public mass transportation if it were available.
3.43 A rotary plug valve needs to be replaced to repair a machine, and the probabilities that the replacement will be a flange style (low pressure), flange style (high pressure),wafer style, or lug style are 0.16, 0.29, 0.26, and 0.15. Find the probabilities that the replacement will be
(a) a flange-style plug;
(b) a flange- (low pressure) or a wafer-style plug;
(c) a wafer-style or a lug-style plug;
(d) a flange-style (high pressure) or a wafer-style or a lug-style plug.
3.44 The probabilities that a TV station will receive 0, 1, 2, 3, . . . , 8 or at least 9 complaints after showing a controversial program are, respectively,
0.01, 0.03, 0.07, 0.15, 0.19, 0.18, 0.14, 0.12, 0.09, and 0.02. What are the probabilities that after showing such a program the station will receive
(a) at most 4 complaints;
(b) at least 6 complaints;
(c) from 5 to 8 complaints?
3.45 If each point of the sample space of Figure 3.12 repre-sents an outcome having the probability 1
32, find Figure 3.12 Diagram for Exercise 3.45
3.46 The probability that a turbine will have a defective coil is 0.10, the probability that it will have defective blades is 0.15, and the probability that it will have both defects is 0.04.
(a) What is the probability that a turbine will have one of these defects?
(b) What is the probability that a turbine will have neither of these defects?
3.47 The probability that a construction company will get the tender for constructing a flyover is 0.33, the prob-ability that it will get the tender for constructing an underpass is 0.28, and the probability that it will get both tenders is 0.13.
(a) What is the probability that it will get at least one tender?
(b) What is the probability that it will get neither tender?
3.49 It can be shown that for any three events A, B, and C, the probability that at least one of them will occur is given by
P( A∪ B ∪ C ) = P( A ) + P( B ) + P(C )
− P( A ∩ B ) − P( A ∩ C )
− P( B ∩ C ) + P( A ∩ B ∩ C ) Verify that this formula holds for the probabilities of Figure 3.13. Figure 3.13 Diagram for Exercise 3.49
3.50 Suppose that in the maintenance of a large medical-records file for insurance purposes the probability of an error in processing is 0.0010, the probability of an error in filing is 0.0009, the probability of an error in retrieving is 0.0012, the probability of an error in pro-cessing as well as filing is 0.0002, the probability of an error in processing as well as retrieving is 0.0003, the probability of an error in filing as well as retrieving is 0.0003, and the probability of an error in processing and filing as well as retrieving is 0.0001. What is the probability of making at least one of these errors?
3.51 If the probability of event A is p, then the odds that it will occur are given by the ratio of p to 1− p. Odds are usually given as a ratio of two positive integers having no common factor, and if an event is more likely not to occur than to occur, it is customary to give the odds that it will not occur rather than the odds that it will occur. What are the odds for or against the occurrence of an event if its probability is
(a) 4
7; (b) 0.05; (c) 0.80?
3.52 Use the definition of Exercise 3.51 to show that if the odds for the occurrence of event A are a to b, where a and b are positive integers, then
p= a
a+ b
3.53 The formula of Exercise 3.52 is often used to deter-mine subjective probabilities. For instance, if an appli-cant for a job “feels” that the odds are 7 to 4 of getting the job, the subjective probability the applicant assigns to getting the job is
p= 7
7+ 4 = 7 11
(a) If a businessperson feels that the odds are 3 to 2 that a new venture will succeed (say, by betting
$300 against $200 that it will succeed), what sub-jective probability is he or she assigning to its success?
(b) If a student is willing to bet $30 against $10, but not $40 against $10, that he or she will get a pass-ing grade in a certain course, what does this tell us about the subjective probability the student as-signs to getting a passing grade in the course?
3.54 Subjective probabilities may or may not satisfy the third axiom of probability. When they do, we say that they are consistent; when they do not, they ought not to be taken too seriously.
(a) The supplier of delicate optical equipment feels that the odds are 7 to 5 against a shipment arriv-ing late, and 11 to 1 against it not arrivarriv-ing at all.
Furthermore, he feels that there is a 50/50 chance (the odds are 1 to 1) that such a shipment will either arrive late or not at all. Are the correspond-ing probabilities consistent?
(b) There are two Ferraris in a race, and an expert feels that the odds against their winning are, respec-tively, 2 to 1 and 3 to 1. Furthermore, she claims that there is a less-than-even chance that either of the two Ferraris will win. Discuss the consistency of these claims.