Unformatted text preview: 82 Chapter 2 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. 2.12.
2.13. 2.14. 2.15. Basic Concepts of Probability Theory (b) Suppose that only the number, and not the sequence, of pens tested in part a is noted
Specify the sample space.
(c) Suppose that the pens are selected one by one and tested until both good pens hav
been identified, and the sequence of test results is noted. What is the sample Space. _ (d) Specify the sample space in part c if only the number of pens tested is noted. Three friends (Al, Bob, and Chris) put their names in a hat and each draws a name from ;
the hat. (Assume Al picks first, then Bob, then Chris.) ’ (a) Find the sample space. (b) Find the sets A, B, and C that correspond to the events “Al draws his name,” “Bob,
draws his name,” and “Chris draws his name.” (c) Find the set corresponding to the event, “no one draws his own name.” ((1) Find the set corresponding to the event, “everyone draws his own name.” (c) Find the set corresponding to the event, “one or more draws his own name.” Let M be the number of message transmissions in Experiment E6. (a) What is the set A corresponding to the event “M is even”? (b) What is the set B corresponding to the event “M is a multiple of 3”? (c) What is the set C corresponding to the event “‘6 or fewer transmissions are re
quired”? ((1) Find the sets A O B, A e B, A F) B H C and describe the corresponding events in
words. A number U is selected at random from the unit interval. Let the events A and B be:
A = “U differs from 1/2 by more than 1/4” and B = “1 — U is less than 1/2.” Find the
events A F) B, At) B, A U B.
The sample space of an experiment is the real line. Let the events A and B correspond to
the following subsets of the real line: A z (#00, r] and B = (#00, s], where r S 3. Find
an expression for the event C 2 (r, s] in terms of A and B. Show that B = A U C and A f) C = Q. ,
Use Venn diagrams to verify the set identities given in Eqs. (2.2) and (2.3). You will need
to use different colors or different shadings to denote the various regions clearly. Show that: (a) If event A implies B, and B implies C, then A implies C.
(b) If event A implies B, then 3" implies A“.
ShowthatifAU B = Aand AﬂB = A'then A = B. Let A and B be events. Find an expression for the event “exactly one of the events A and
B occurs.” Draw a Venn diagram for this event. Let A, B, and C be events. Find expressions for the following events:
(a) Exactly one of the three events occurs. (b) Exactly two of the events occur. (c) One or more of the events occur. ((1) Two or more of the events occur. (e) None of the events occur. Figure P2.1 shows three systems of three components, C1, C2, and C3. Figure P2.1(a) iSa
“series” system in which the system is functioning only if all three components are func 1.” Problems 83 system in which the system is functioning as long as at least two components are fune—
tionmg. Let A k be the event “component k is functioning.” For each of the three system
configurations, express the event “system is functioning” in terms of the events A k .. (a) Series system (b) Parallel system (c) Twoout—of—three system FlGURE P2.1 2.16. A system has two key subsystems. The system is “up” if both of its subsystems are fune tioning..Triple redundant systems are configured to provide high reliability. The overall
system Is operational as long as one of three systems is “up.” Let A jk correspond to the
event “unit k in system j is functioning,” for j = 1, 2, 3 and k = 1, 2. . (a) Write an expression for the event “overall system is up.” (1)) Explain why the above problem is equivalent to the problem of having a connection
in the network of switches shown in Fig. P22. A11 A12 FIGURE P22 2.17. In a specified 6AM—tO—6AM 24—hour period, a student wakes up at time t1 and goes to
sleep at some later time 1‘2. (a) Find the sample space and sketch it on the x—y plane if the outcome of this experi
ment consists of the pair (11, t2). (b) Specify the set A and sketch the region on the plane corresponding to the event “stu—
dent is asleep at noon.” (c) Specify the set B and sketch the region on the plane corresponding to the event “stu—
dent sleeps through breakfast (7—9 AM)” In n1 ...
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