28 CHAPTER 2
(f) The set A∪Bcontains the outcomes that are either in A, in B, or in both. Therefore A∪B={CCC,
DDD, SSS, CDS, CSD, DCS, DSC, SCD, SDC}.
5. (a) The outcomes are the sequences of candidates that end with either #1 or #2. These are {1, 2, 31,
32, 41, 42, 341, 342, 431, 432}.
6. (a) The equally likely outcomes are the sequences of two distinct candidates. These are {12, 13, 14, 21,
23, 24, 31, 32, 34, 41, 42, 43}.
Page 28
SECTION 2.1 29
8. (a) 0.7
9. (a) The events of having a major flaw and of having only minor flaws are mutually exclusive. Therefore
10. (a) False
11. (a) False
Page 29
30 CHAPTER 2
13. (a) P(S∪C) = P(S) + P(C)−P(S∩C)
14. (a) Since 562 stones were neither cracked nor discolored, 38 stones were cracked, discolored, or both. The
15. (a) Let Rbe the event that a student is proficient in reading, and let Mbe the event that a student is
SECTION 2.1 31
(b) We need to find P(R∩Mc). Now P(R) = P(R∩M) + P(R∩Mc) (this can be seen from a Venn
(c) First we compute P(R∪M):
17. P(A∩B) = P(A) + P(B)−P(A∪B)
(b) P(does not contain B) = 1 −P(contains B)
32 CHAPTER 2
20. (a) Aand Bare mutually exclusive, since it is impossible for both events to occur.
Section 2.2
1. (a) (4)(4)(4) = 64
Page 32
5. (a) (8)(7)(6) = 336
7. (210)(45) = 1,048,576
Page 33
10. 15
6,5,4=15!
6!5!4! = 630,630
Section 2.3
1. Aand Bare independent if P(A∩B) = P(A)P(B). Therefore P(B) = 0.25.
3. (a) 5/15
Page 34
SECTION 2.3 35
(b) P(2 resistors selected) = P(1st is 50Ω and 2nd is 100Ω)
5. Given that a student is an engineering major, it is almost certain that the student took a calculus
7. Let Arepresent the event that the biotechnology company is profitable, and let Brepresent the event
that the information technology company is profitable. Then P(A) = 0.2 and P(B) = 0.15.
8. Let Mdenote the event that the main parachute deploys, and let Bdenote the event that backup
9. Let Vdenote the event that a person buys a hybrid vehicle, and let Tdenote the event that a person
buys a hybrid truck. Then
10. Let Adenote the event that the allocation sector is damaged, and let Ndenote the event that a
non-allocation sector is damaged. Then P(A∩Nc) = 0.20, P(Ac∩N) = 0.7, and P(A∩N) = 0.10.
SECTION 2.3 37
(d) P(A|N) = P(A∩N)
(f) P(Ac|N) = P(Ac∩N)
P(N)
Page 37
11. Let OK denote the event that a valve meets the specification, let Rdenote the event that a valve is
(e) P(OK) = P(OK ∩Rc) + P(OK ∩R)
Page 38
SECTION 2.3 39
12. Let Sdenote Sarah’s score, and let Tdenote Thomas’s score.
13. Let T1 denote the event that the first device is triggered, and let T2 denote the event that the second
device is triggered. Then P(T1) = 0.9 and P(T2) = 0.8.
(b) P(T1c∩T2c) = P(T1c)P(T2c) = (1 −0.9)(1 −0.8) = 0.02
Page 39
14. Let Ldenote the event that Laura hits the target, and let P h be the event that Philip hits the target.
Then P(L) = 0.5 and P(P h) = 0.3.
= 0.65
= 0.5
(c) P(L|exactly one hit) = P(L∩exactly one hit)
P(exactly one hit)
15. (a) 88
88 + 12 = 0.88
Page 40
SECTION 2.3 41
17. (a) 56 + 24
100 = 0.80
18. (a) 71
102 + 71 + 33 + 134 =71
340
Page 41
19. Let R,D, and Idenote the events that the senator is a Republian, Democrat, or Independent,
respectively, and let Mand Fdenote the events that the senator is male or female, respectively.
= 0.53
20. Let Gdenote the event that a customer is a good risk, let Mdenote the event that a customer
is a medium risk, let Pdenote the event that a customer is a poor risk, and let Cbe the event
that a customer has filed a claim. Then P(G) = 0.7, P(M) = 0.2, P(P) = 0.1, P(C|G) = 0.005,
Page 42
SECTION 2.3 43
21. (a) That the gauges fail independently.
22. No. P(both gauges fail) = P(first gauge fails)P(second gauge fails|first gauge fails).
23. (a) P(A) = 3/10
Page 43
44 CHAPTER 2
24. (a) P(A) = 300/1000 = 3/10
(b) Given that Aoccurs, there are 999 components remaining, of which 299 are defective.
26. Let Edenote the event that a parcel is sent express (so Ecdenotes the event that a parcel is sent
Page 44
SECTION 2.3 45
27. Let Rdenote the event of a rainy day, and let Cdenote the event that the forecast is correct. Then
28. Let Adenote the event that the flaw is found by the first inspector, and let Bdenote the event that
the flaw is found by the second inspector.
29. Let Fdenote the event that an item has a flaw. Let Adenote the event that a flaw is detected by the
first inspector, and let Bdenote the event that the flaw is detected by the second inspector.
46 CHAPTER 2
(a) P(F|Ac) = P(Ac|F)P(F)
P(Ac|F)P(F) + P(Ac|Fc)P(Fc)
30. Let Ddenote the event that a person has the disease, and let + denote the event that the test is
positive. Then P(D) = 0.05, P(+|D) = 0.99, and P(+|Dc) = 0.01.
31. (a) Each child has probability 0.25 of having the disease. Since the children are independent, the proba-
bility that both are disease-free is 0.752= 0.5625.
Page 46
SECTION 2.3 47
(b) Each child has probability 0.5 of being a carrier. Since the children are independent, the probability
32. Let F l denote the event that a bottle has a flaw. Let Fdenote the event that a bottle fails inspection.
We are given P(F l) = 0.0002, P(F|F l) = 0.995, and P(Fc|F lc) = 0.99.
Page 47