Question

A commercial jet aircraft has two engines. Both engines have a reliability of 0.90, in other words the probability that a given engine will not fail is 0.90. Engines are assumed to operate independently from each other.

a. Do you think the following two events are mutually exclusive?

Event 1: Engine 1 will not fail
Event 2: Engine 2 will not fail

A) Yes, since they are independent

B) No

C) There is not enough information to determine if they are

D)Yes, since they are dependent   

b.What is the probability that both engines will fail?

c.Given that the second engine has failed, what is the probability that the first engine will fail?

d. Given that the first engine has failed, what is the probability that the second engine will fail?

e.What is the probability that neither engine will fail?

f.What is the probability that at least one of the engines will fail?

g.What is the probability that exactly one engine will fail?

Answer 1

a. B) No

The events are not mutually exclusive, since if Engine 1 does not fail it doesn't ensure that Engine 2 will also not fail or vice versa because the two engines are assumed to operate independently from each other.

b. P[both engines will fail]

= 0.10 x 0.10

= 0.01

c. P[First engine will fail | Second engine has failed]

= P[First engine will fail    Second engine has failed] / P[ second engine has failed]

= P[First engine will fail] . P[ Second engine has failed]/P[ second engine has failed]

= P[First engine will fail]

= 0.10

d.

P[Second engine will fail | First engine has failed]

= P[second engine will fail   first engine has failed] / P[ first engine has failed]

= P[second engine will fail] . P[ first engine has failed]/P[ first engine has failed]

= P[second engine will fail]

= 0.10

e. P[neither engine will fail]

= 0.9 x 0.9

= 0.81

f. P[ at least one engine will fail]

= 1 - P[no engine will fail]

= 1 - 0.81

= 0.19

g. P[exactly one engine will fail]

= 1 - P[no engine will fail] - P[both engines fail]

= 1 - 0.81 - 0.01

= 0.18