HW#3 Solution
5.16.
The probabilities of independent events A and B must satisfy the equation P(A and B) = P(A)P(B). If A and
B were also mutually exclusive, then P(A and B) would equal 0, which would mean that P(A)P(B) = P(A
and B) = 0. This would require that at least one of A or B have zero probability of occurring. Although this
is technically possible, most events of interest have non-zero probabilities, making P(A)P(B) non-zero. It
is therefore impossible for independent events with non-zero probabilities to be mutually exclusive.
5.19.
(a) P(both bids are successful) = P(E1 and E2) = P(E1)P(E2) = (.4)(.3) = .12.
(b) P(neither bid is successful) = P(E1 and E2) = P(E1)P(E2) = (1-.4)(1-.3) = .42.
Recall that if two events are independent, so are their complements.
(c) P(at least one bid is successful) = 1 - P(neither is successful) = 1 - P(E1 and E2)
= 1 - .42 = .58.
5.20.
Let A denote the event that components 3 and 4 both work correctly and let B denote the event that at
least one of components 1 or 2 works correctly. Then P(systems works) = P(A or B). From the general
addition law, P(A or B) = P(A) +P(B) - P(A and B). Because all components act independently of one
another, P(A) = P(3 and 4 work) = P(3 works)P(4 works) = (.9)(.9) = .81. P(B) = P(1 or 2 works) = P(1 works)
+ P(2 works) - P(1 and 2 work) = .9 + .9 - (.9)(.9) = .99. Finally, events A and B are independent since A
involves only components 3 and 4, whose actions are independent of components 1 and 2, so P(A and B) =
P(A)P(B) = (.81)( .99) = .8019. Therefore, P(A or B) = P(A) +P(B) - P(A and B) = .81 + .99 - .8019 = .9981.
4
(a) Let A denote the event that the next customer requests extra unleaded gas.
Let B denote the event that the next customer fills the tank.
What we’ve know:
35% customers use extra gas. It’s P(A)=0.35
60% will fill the tank, given of the customers using extra gas. It’s P(B|A)= 0.6
P(A and B) = P(B|A)P(A)=0.35*0.6 = 0.21
(b) We’ve known P(B) = 0.455. We want to know P(A|B).
P(A|B) = P(A and B) / P(B) = 0.21/0.455= 0.461