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Basic probability aeroplane engines problem!

  1. May 11, 2014 #1
    Hi all, I've got a past paper question that goes something like this:

    An aeroplane has four engines. During a certain journey, each engine fails with a probability of 0.1, independantly of the others. The aeroplane can fly when at least two engines are working. Calculate:

    A) The probability that the aeroplane will complete the journey.
    B) The probability that the aeroplane will complete four journeys with no engine failures. State clearly any assumptions you make.


    Ok so I've decided this is a binomial distribution problem; for part A) I have assigned X to be the r.v. "number of engines failing" where X~B(4,0.1).

    I'm looking for Pr(X ≤ 2) = 1 - (Pr(X = 3) + Pr(X = 4))
    = 1 - (4C3X0.1^3X0.9 + 4C4X0.1^4)
    = 1 - 3.7 x 10^-3
    = 0.9963 (which as far as answers go, seems relatively plausible?).

    As for part B), I have absolutely no ideas! If someone would be so kind as to assist it'd be much appreciated :)

    Cheers,

    Dan
     
  2. jcsd
  3. May 11, 2014 #2

    AlephZero

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    Start by finding the probability of one journey with no engine failures.
     
  4. May 11, 2014 #3

    Ray Vickson

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    In B): what is the probability that in any single trip none of the engines fail? What would this become for 4 trips?
     
  5. May 11, 2014 #4
    Thanks for the replies guys. So, for one journey the probability that no engines fail is given by:

    Pr(X = 0) = 4C0X0.9^4 = 0.6561

    And so for four journeys, 0.6561^4 = 0.185? This seems a little bit of an unrealistic figure?
     
  6. May 11, 2014 #5

    AlephZero

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    Yes, because the assumption that the engine failure rate per flight os 0.1 is unrealistic. The actual rate for modern jet engines is of the order of 1 failure per 100,000 hours. Even for 10 hour long haul flights, that is a probability of about 0.0001 per flight, not 0.1.
     
  7. May 11, 2014 #6
    Hi Aleph, thanks for the clarification!
     
  8. May 11, 2014 #7

    Ray Vickson

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    I echo the remarks of AlephZero: it is the 0.1 figure that is unrealistic. Would you want to fly in a plane that had only a 99.6% probability of not crashing? But, given p = 0.1 the answer you got is understandable: it is like having a 16-engine plane and asking that none of the engines fail.
     
  9. May 11, 2014 #8

    AlephZero

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    That general principle applies in real life to 2 and 4 engined planes. Other things being equal, you are twice as likely to have an engine failure with 4 engines than with 2. But the consequence of 1 engine failure out of 4 is likely to be less serious than 1 out of 2, so it's not a clear-cut decision whether 2 or 4 engines is "best" for all plane designs.

    Also, the question assumes the engine failures are independent of each other, and that is not necessarily true either. Running out of fuel, or flying into a flock of birds big enough to cause engine damage, are two examples where it is false.
     
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