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Math Help - Probability Problem

  1. #1
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    Probability Problem

    A law firm employs three types of lawyers: junior lawyers, senior lawyers, and partners. During a given year, there is a 0.15 prabability that a junior lawyer will be promoted to senior lawyer and a 0.05 probability that he or
    she will leave the firm. Also, there is a 0.20 probability that a senior lawyer will be promoted to partner and a 0.10 proability that he or she will leave the firm. There is a 0.05 proability that a partner will leave the firm. The firm never demotes a lawyer.
    (1) How long does a newly hired junior lawyer stay
    with the firm?
    (2) What is the probability that a newly hired
    junior lawyer will leave the firm as a partner?
    (3) What is the
    average length of time that a partner spends with the firm (as
    the partner)?
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  2. #2
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    Quote Originally Posted by shark View Post
    A law firm employs three types of lawyers: junior lawyers, senior lawyers, and partners. During a given year, there is a 0.15 prabability that a junior lawyer will be promoted to senior lawyer and a 0.05 probability that he or

    she will leave the firm. Also, there is a 0.20 probability that a senior lawyer will be promoted to partner and a 0.10 proability that he or she will leave the firm. There is a 0.05 proability that a partner will leave the firm. The firm never demotes a lawyer.

    [snip]

    (3) What is the
    average length of time that a partner spends with the firm (as

    the partner)?
    Let X be the random variable number of years the partner stays at the firm.
    X ~ Geometric(p = 0.05).
    E(X) = 1/p = 20.
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  3. #3
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    Quote Originally Posted by shark View Post
    [snip]

    (2) What is the probability that a newly hired
    junior lawyer will leave the firm as a partner?
    [snip]
    = Pr(junior lawyer becomes senior lawyer) Pr(Senior lawyer becomes partner) Pr(Partner leaves firm).

    Pr(junior lawyer becomes senior lawyer) = 0.15 + (0.8)(0.15) + (0.8)^2 (0.15) + ......

    = 0.15 (1 + 0.8 + 0.8^2 + ....) = 0.15 \left( \frac{1}{1 - 0.8} \right) = \frac{3}{4}.

    Do the other bits the same way .....
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  4. #4
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    Quote Originally Posted by shark View Post
    [snip](1) How long does a newly hired junior lawyer stay
    with the firm?
    [snip]
    When the junior lawyer leaves, s/he can leave either as a junior lawyer, a senior lawyer or a partner ....
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  5. #5
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    Quote Originally Posted by shark View Post
    A law firm employs three types of lawyers: junior lawyers, senior lawyers, and partners. During a given year, there is a 0.15 prabability that a junior lawyer will be promoted to senior lawyer and a 0.05 probability that he or

    she will leave the firm. Also, there is a 0.20 probability that a senior lawyer will be promoted to partner and a 0.10 proability that he or she will leave the firm. There is a 0.05 proability that a partner will leave the firm. The firm never demotes a lawyer.
    (1) How long does a newly hired junior lawyer stay
    with the firm?
    [snip]
    Quote Originally Posted by mr fantastic View Post
    When the junior lawyer leaves, s/he can leave either as a junior lawyer, a senior lawyer or a partner ....
    By the way, this is an ideal question to apply the technique of Monte Carlo simulation ....
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