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    Tony Kelly & Kieran Mills) : Aths Roblem O

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    ishaan1589

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    Tony Kelly & Kieran Mills) : Aths Roblem O

    Uploaded by

    ishaan1589
    3 views3 pages

    Original Title

    Problem 2

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    Copyright:

    Attribution Non-Commercial (BY-NC)
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    Download as pdf or txt
    3 views3 pages

    Tony Kelly & Kieran Mills) : Aths Roblem O

    Uploaded by

    ishaan1589

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    MATHS PROBLEM NO.

    2 ( Tony Kelly & Kieran Mills)

    MATHS PROBLEM NO. 2 An aeroplane engine fails with probability x. A three-engined plane needs 2 or 3 engines for a successful flight. Find (i) the probability of a successful flight on a one-engined plane in terms of x, (ii) the probability of a successful flight in a three-engined plane,

    (iii) the range of values of x for which a one-engined plane is safer than a three-engined plane.

    MATHS PROBLEM NO. 2 ( Tony Kelly & Kieran Mills)

    MATHS PROBLEM NO. 2 An aeroplane engine fails with probability x. A three-engined plane needs 2 or 3 engines for a successful flight. Find (i) the probability of a successful flight on a one-engined plane in terms of x, (ii) the probability of a successful flight in a three-engined plane,

    (iii) the range of values of x for which a one-engined plane is safer than a three-engined plane. SOLUTION (i) One engined plane: p(Failed flight) = x p(Successful flight) = (1 x) (ii) Three-engined flight: p(Successful flight) = p(3 engines working) or p(2 engines working and one failing)

    = (1 x) (1 x) (1 x) + (1 x) (1 x) x = (1 x)(1 x)(1 x) + 3 x(1 x)(1 x) = (1 x) 2 [(1 x) + 3 x] = (1 x) 2 [1 x + 3 x] = (1 x) 2 [2 x + 1]

    3! 2!

    MATHS PROBLEM NO. 2 ( Tony Kelly & Kieran Mills)

    MATHS PROBLEM NO. 2 An aeroplane engine fails with probability x. A three-engined plane needs 2 or 3 engines for a successful flight. Find (i) the probability of a successful flight on a one-engined plane in terms of x, (ii) the probability of a successful flight in a three-engined plane,

    (iii) the range of values of x for which a one-engined plane is safer than a three-engined plane. SOLUTION (iii) p(Successful flight in a one-engined plane) > p(Successful flight in a three-engine flight)
    (1 x) > (1 x) 2 (2 x + 1) (1 x) (1 x) 2 (2 x + 1) > 0 (1 x)[1 (1 x)(2 x + 1)] > 0 (1 x)[1 (2 x + 1 2 x 2 x)] > 0 (1 x)[1 ( x + 1 2 x )] > 0
    2

    Solve the equality: x(1 x)(2 x 1) = 0 x = 0, 1,


    1 2

    NOTE: There are no solutions less than 0 or greater than 1 as the probability of an event must be between 0 and 1.
    1 2

    x=

    1 4

    x=

    3 4

    (1 x)[1 x 1 + 2 x 2 ] > 0 (1 x)[2 x 2 x] > 0 (1 x)[ x(2 x 1)] > 0 x(1 x)(2 x 1) > 0

    1 1 (1 4 )(1 4 )( 2( 4 ) 1) 3 1 = (1 4 )( 4 )( 2 1) 3 1 = (1 4 )( 4 )( 2 )

    3 3 (3 4 )(1 4 )( 2( 4 ) 1) 3 1 = (3 4 )( 4 )( 2 1) 1 1 = (3 4 )( 4 )( 2 )

    <0 False

    >0 True

    ANSWER:

    1 2

    < x <1

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