Smile 發問於 科學及數學數學 · 4 月前

Maths problem: how to do, thanks?

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    4 月前
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    Question:

    A professional qualification examination consists of two papers, I and II. To obtain the qualification successfully, a candidate has to pass both papers. If the candidate fails in any one of the papers, he/she needs to retake the paper again. It is given that the probabilities of Julia passing the two papers are 7/9 and 2/3 respectively.

    (a)

    Find the probabilities that Julia obtains the professional qualification

    (i) by sitting each paper once,

    (ii) by retaking one paper once.

    (b)

    It is given that the probabilities of Tommy passing the two papers are 5/7 and 3/4 respectively. If both Tommy and Julia attend the examination, find the probabilities that

    (i) at least one of them obtains the professional qualification without retake.

    (ii) each of them obtains the professional qualification by taking one paper.

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    Solution:

    (ai)

    P(Julia obtains the professional qualification by sitting each paper once)

    = P(Pass paper I) P(Pass paper II)

    = (7/9) (2/3)

    = 14/27

    (aii)

    P(Julia obtains the professional qualification by retaking one paper once)

    = P(Pass paper I) P(Fail in paper II) P(Pass paper II)

       + P(Fail in paper I) P(Pass paper II) P(Pass paper I)

    = (7/9) (1 - 2/3) (2/3) + (1 - 7/9) (2/3) (7/9)

    = 70/243

    (bi)

    Let

    A be the event A that Julia cannot obtains the professional qualification without retake,

    B be the event B that Tommy cannot obtains the professional qualification without retake.

    P(A)

    = 1 - P(Julia obtains the professional qualification by sitting each paper once)

    = 1 - 14/27 ...... (ai)

    = 13/27

    P(B)

    = 1 - P(Tommy obtains the professional qualification by sitting each paper once)

    = 1 - P(Tommy passes paper I) P(Tommy passes paper II)

    = 1 - (5/7) (3/4)

    = 13/28

    P(at least one of them obtains the professional qualification without retake)

    = 1 - P(None of them obtains the professional qualification without retake)

    = 1 - P(A) P(B)

    = 1 - (13/27) (13/28)

    = 587/756

    (bii)

    P(Tommy obtains the professional qualification by retaking one paper once)

    = P(Tommy passes paper I) P(Tommy fails in paper II) P(Tommy passes paper II)

       + P(Tommy fails in paper I) P(Tommy passes paper II) P(Tommy passes paper I)

    = (5/7) (1 - 3/4) (3/4) + (1 - 5/7) (3/4) (5/7)

    = 225/784

    P(Each of them obtains the professional qualification by taking one paper)

    = P(Julia obtains the professional qualification by retaking one paper once)

       × P(Tommy obtains the professional qualification by retaking one paper once)

    = (70/243) (225/784)

    = 125/1512

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