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# Advanced Math/Probability of independent events

Question
A tri motor plane has three engines – a central engine and one on each wing. The plane will only crash if the central engine fails AND one of the wing engines also fails. The probability of failure during any given flight is 0.05 for the central engine and 0.08 for each of the wing engines. Assuming the engines operate independently, determine the probability of the plane crashing during a flight.

I just can't agree with my college teacher for the correct answer. He is for 2(0.08*0.05)+0.05*0.08^2, because the 3 engine failure is independent.
And i'm for 2(0.08*0.05)-0.05*0.08^2, because the 3 engine failure
is accounted for twice in the first multiplication.
I would be very happy if you could answer this question!

Country:   Sligo, Ireland
Private:   No
Subject:   math probability
Question:   A tri motor plane has three engines – a central engine and one on each wing. The plane will only crash if the central engine fails AND one of the wing engines also fails. The probability of failure during any given flight is 0.05 for the central engine and 0.08 for each of the wing engines. Assuming the engines operate independently, determine the probability of the plane crashing during a flight.

I just can't agree with my college teacher for the correct answer. He is for 2(0.08*0.05)+0.05*0.08^2, because the 3 engine failure is independent.
And i'm for 2(0.08*0.05)-0.05*0.08^2, because the 3 engine failure
is accounted for twice in the first multiplication.
I would be very happy if you could answer this question!
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Let:

C = "central engine fails"
L = "leftwing engine fails"
R = "righwing engine fails"

p(C) = 0.05
p(L) = p(R) = 0.08  <<< those wing engines are probably made in China, right?

Then

p(crash) = p(C and [L or R])

Now, for independent events, the rules are:

P(A and B) = p(A) p(b)
p(A or B) = p(A) + p(b) - p(A and B)
= p(A) + p(b) - p(A) p(B)

p(crash) = p(C and [L or R])

= p(C) p(L or R)

= p(C)[p(L) + p(R) - p(L) p(R)]

= 0.05 [0.08 + 0.08 - 0.08^2]

= 2(0.05)(0.08) - 0.05(0.08)^2

I think I'm with you.

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#### Paul Klarreich

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