Within a company there are three grades of employees. Yearly reviews take place

Question

Within a company there are three grades of employees. Yearly reviews take place at which time participants are faced with one of four outcomes:

Promoted,

Remain in current grade,

Demoted,

Quit (causing a vacancy that is filled at Grade 1)

The probability of getting promoted from Grade 1 to Grade 2 is 0.6 while the probability of someone at Grade 1 quitting is 0.3. For Grade 2 employees the probability of being promoted is 0.3 while the probability of remaining at the current grade is 0.5. It is equally likely that you might quit or get demoted if you are at Grade 2 whereas the probability of getting demoted from Grade 3 is the same as quitting and is 0.1. The remainder of the employees remain at Grade 3.

Outline the method to how the long-term probability that a person makes it to, and remains, in Grade 3. (There is no need to solve for the probabilities explicitly - i.e., stop at deriving the equations).

Explanation / Answer

Solution

There are 5 ways of getting to Grade 3

1). Get promoted from Grade 1 to Grade 2 and then subsequently get promoted from Grade 2 to Grade 3 – probability for this = 0.6 x 0.3 …………………………………………………….(1)

2). Get promoted from Grade 2 to Grade 3 – probability for this = 0.3 ……………………….(2)

3). Get demoted from Grade 3 to Grade 2, but subsequently get promoted from Grade 2 to Grade 3 – probability for this = 0.1 x 0.6 x 0.3 …….………………………………………….(3)

4). Get demoted from Grade 3 to Grade 2 and to Grade 1, but subsequently get promoted from Grade 1 to Grade 2 and then to Grade 3 – probability for this = 0.1 x 0.1 x 0.6 x 0.3 ……….(4)

5). Get demoted from Grade 2 to Grade 1, but subsequently get promoted from Grade 1 to Grade 2 and then to Grade 3 – probability for this = 0.1 x 0.6 x 0.3 …………………….….(5)

Thus total probability of reaching Grade 3 = (1) + (2) + (3) + (4) + (5) ……………………(6)

Having reached Grade 3, probability of remaining there

= 1- P(demotion to Grade 2) - P(quit) = 0.8 …………………………………………………(7)

Thus, the long-term probability that a person makes it to, and remains, in Grade 3,

= (6) x (7)

= {(0.6x0.3) + (0.3) + (0.1x0.6x0.3) + (0.1x0.1x0.6x0.3) + (0.1x0.6x0.3)}(0.8) ANSWER

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