Question

1.An airoplane with four engines is able to fly if at least one engine is functioning on each wing. Let Ai be the event that the ith engine is working, i = 1, 2, 3, 4, where engines 1 and 2 are on the left wing and engines 3 and 4 are on the right wing. Let A be the event that the airoplane is able to fly. Express A in terms of A1, . . . , A4, using complements, intersections and/or unions.

2. Consider the airplane from question 1. Assume that all engines have a probability of 0.1 of failing, and that they fail independently of each other. What is the probability that the plane will be able to fly?

Answer 1

1.

Left Wing : A1, A2

Right Wing : A3,A4

L : Left Wing Function : At least one Engine on Left Wing : (A1 and A2) OR (A1 and Not A2) OR (Not A1 And A2)

i.e

Similarly

Right Wing Function : At least one Engine on Right Wing : (A3 and A4) OR (A3 and Not A4) OR (Not A3 And A4)

i.e

Let A be the event that the airoplane is able to fly if Both left wing and right wing functioning

i.e L and R =

in another words

2. Probability of a engine failing =0.1

i.e Probability of a engine not failing =1-0.1 =0.9

Probability that the plane will be able to fly = Probability that (Left wing Functions and Right wing functions)

= P(L R) = P(L) P(R)

Probability that left wing functions :P(L)

= Probability that at least one of A1 and A2 is not failing

= 1 - (Probability that both A1 and A2 fail )

Probability that both A1 and A2 fail

Probability that both A1 and A2 fail = 0.01

Probability that left wing functions :P(L)

= Probability that at least one of A1 and A2 is not failing = 1-0.01 = 0.99

Probability that left wing functions :P(L) =0.99

Similarly ,

Probability that right wing functions :P(R)

= Probability that at least one of A3and A4 is not failing

= 1 - (Probability that both A3 and A4 fail )

Probability that both A3 and A4 fail

Probability that both A3 and A4 fail = 0.01

Probability that right wing functions :P(R)

= Probability that at least one of A3 and A4 is not failing = 1-0.01 = 0.99

Probability that right wing functions :P(R) =0.99

P(L R) = P(L) P(R) = 0.99 x 0.99 = 0.9801

Probability that (Left wing Functions and Right wing functions) = P(L R) = 0.9801

Probability that the plane will be able to fly = 0.9801