Page 50 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 33

2.23 Events A and B are mutually exclusive. Determine which of the following relations
are true and which are false:
a) P AjB) ˆ P A).
b) P A [ BjC) ˆ P AjC) ‡ P BjC).
c) P A) ˆ 0, P B) ˆ 0, or both.
P AjB) P BjA)
d) ˆ .
P B) P A)
e) P AB) ˆ P A)P B).
Repeat the above if events A and B are independent.
2.24 On a stretch of highway, the probability of an accident due to human error in any

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given minute is 10 , and the probability of an accident due to mechanical break-
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down in any given minute is 10 . Assuming that these two causes are independent:
(a) Find the probability of the occurrence of an accident on this stretch of highway
during any minute.
(b) In this case, can the above answer be approximated by P(accident due to
human error) ‡ P(accident due to mechanical failure)? Explain.
(c) If the events in succeeding minutes are mutually independent, what is the
probability that there will be no accident at this location in a year?
2.25 Rapid transit trains arrive at a given station every five minutes and depart after
stopping at the station for one minute to drop off and pick up passengers. Assum-
ing trains arrive every hour on the hour, what is the probability that a passenger
will be able to board a train immediately if he or she arrives at the station at a
random instant between 7:54 a.m. and 8:06 a.m.?
2.26 A telephone call occurs at random in the interval (0, t). Let T be its time of
occurrence. Determine, where 0 t 0 t 1 t:
(a) P t 0 T t 1 ).
(b) P t 0 T t 1 jT t 0 ).
2.27 For a storm-sewer system, estimates of annual maximum flow rates (AMFR) and
their likelihood of occurrence [assuming that a maximum of 12 cfs (cubic feet per
second) is possible] are given as follows:

Event A ˆ 5 to 10 cfs†; P A†ˆ 0:6:
Event B ˆ 8 to 12 cfs†; P B†ˆ 0:6:
Event C ˆ A [ B; P C†ˆ 0:7:

Determine:
(a) P (8 AMFR 10), the probability that the AMFR is between 8 and 10 cfs.
(b) P (5 AMFR 12).
(c) P (10 AMFR 12).
(d) P (8 AMFR 10j5 AMFR 10).
(e) P (5 AMFR 10jAMFR 5).
2.28 At a major and minor street intersection, one finds that, out of every 100 gaps on
the major street, 65 are acceptable, that is, large enough for a car arriving on the
minor street to cross. When a vehicle arrives on the minor street:
(a) What is the probability that the first gap is not an acceptable one?
(b) What is the probability that the first two gaps are both unacceptable?
(c) The first car has crossed the intersection. What is the probability that the
second will be able to cross at the very next gap?

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