# Probability Problem

• Apr 24th 2008, 03:50 AM
shark
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)?
• Apr 24th 2008, 05:14 AM
mr fantastic
Quote:

Originally Posted by shark
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.
• Apr 25th 2008, 07:44 PM
mr fantastic
Quote:

Originally Posted by shark
[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) $\displaystyle = 0.15 + (0.8)(0.15) + (0.8)^2 (0.15) + ......$

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

Do the other bits the same way .....
• Apr 25th 2008, 07:49 PM
mr fantastic
Quote:

Originally Posted by shark
[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 ....
• Apr 26th 2008, 08:52 PM
mr fantastic
Quote:

Originally Posted by shark
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
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 ....