A contractor has two subcontractors for his excavation work. Experience shows that in 60% of the time, subcontractor A was available to do the job, whereas subcontractor B was available 805 of the time. Also, the contractor is able to get at least one of these two subcontractors 905 of the time. (a) What is the probability that both subcontractors will be available to do the next job? (b) If the contractor learned that subcontractor A is not available for the job, what is the probability that the other subcontractor will be available? (c) Suppose, EA denotes the event that subcontractor A is available and EB donates that subcontractor B is available: (i) Are events EA and EB statistically independent? (ii) Are events EA and EB mutually exclusive? (iii)Are events EA and EB collectively exhaustive? Solution a)P(EA n EB) = P(EA) + P(EB) - P(EA U EB) = .6 + .8 - .9 = .5 b) Using ~ to indicate complement P(EA~ n EB) = P(EB) - P(EA n EB) = .8 - .5 = .3 P(EA~) = 1 - P(EA) = 1 - .6 = .4 Then, P(EB|EA~) = P(EA~ n EB)/P(EA~) = .3/.4 = 3/4 c)i) EA and EB are independent iff P(EA)*P(EB)=P(EA n EB) P(EA) * P(EB) = .8 * .6 = .48 P(EA n EB) = .5 .5 is not equal to .48 EA and EB are not independent. ii) P(EA n EB) = .5 is not equal to 0 EA and EB are not mutually exclusive. iii) Events EA and EB are not collectively exhaustive because P(EA U EB) = .9 is not equal to 1..

1. A contractor has two subcontractors for his excavation work. Experience shows that in 60% of
the time, subcontractor A was available to do the job, whereas subcontractor B was available
805 of the time. Also, the contractor is able to get at least one of these two subcontractors 905 of
the time. (a) What is the probability that both subcontractors will be available to do the next job?
(b) If the contractor learned that subcontractor A is not available for the job, what is the
probability that the other subcontractor will be available? (c) Suppose, EA denotes the event that
subcontractor A is available and EB donates that subcontractor B is available: (i) Are events EA
and EB statistically independent? (ii) Are events EA and EB mutually exclusive? (iii)Are
events EA and EB collectively exhaustive?
Solution
a)P(EA n EB) = P(EA) + P(EB) - P(EA U EB) = .6 + .8 - .9 = .5 b) Using ~ to
indicate complement P(EA~ n EB) = P(EB) - P(EA n EB) = .8 - .5 = .3 P(EA~) = 1 - P(EA) = 1 -
.6 = .4 Then, P(EB|EA~) = P(EA~ n EB)/P(EA~) = .3/.4 = 3/4 c)i) EA and EB are independent
iff P(EA)*P(EB)=P(EA n EB) P(EA) * P(EB) = .8 * .6 = .48 P(EA n EB) = .5 .5 is not equal to
.48 EA and EB are not independent. ii) P(EA n EB) = .5 is not equal to 0 EA and EB are not
mutually exclusive. iii) Events EA and EB are not collectively exhaustive because P(EA U EB)
= .9 is not equal to 1.