Question

 

Suppose you are about to start working for Apple Corp. as an intern, so you do some research on the various job positions within the company. In your area of work, there are 4 basic positions that you can hold: intern, programmer, analyst, and project manager.

Each year there is a chance for promotion, and there is also a chance of “involuntary separation” from the company. (In other words, getting fired.)

If you are an intern, there is a 50% chance each year of being promoted to programmer, and a 10% of being fired.

If you are a programmer, there is a 12% chance each year of being promoted to analyst, and a 6% of being fired.

If you are an analyst, there is a 5% chance each year of being promoted to project manager, and a 3% of being fired.

If you are a project manager, you cannot be promoted but there is a 5% chance of being fired.

If you are not promoted or fired, then you stay in the same position. If you are fired, then you cannot be rehired.

What are the different states in this system?

Using excel, construct the transition matrix T for this system. Make sure to label the states.

Assuming you start working as an intern, what is the initial-state matrix S₀?

Find S_5, S₁₀ and S₂₀ for this system.

What is the probability that you will be an intern after 10 years?

What is the probability that you will be a project manger in 5 years?

What is the probability that you will be fired within 20 years?

Answer 1

It would be most logical to assign state 0 to getting fired, and the other 4 sequential states from 1 through 4. The transition matrix T can be obtained by first writing the transition probability towards promotion and being fired using given numerical values, and then the probability to stay in the same state as 1 minus the sum of those probabilities. Hence, we have the states and transition matrix

The initial state matrix is a row vector that represents the quantities in each state at the beginning. Since it is straightforward stated that you start as an intern, hence the required matrix is

Now we know that given the initial state matrix S₀ and the transition matrix T, the state matrix after n steps is given by

To begin with, let us compute all required powers of T. Reaching to the 5^th power takes three steps, after that it will be straightforward squaring. The excel work using MMULT() function is obtained as

T=	1	0	0	0	0
	0.1	0.4	0.5	0	0
	0.06	0	0.82	0.12	0
	0.03	0	0	0.92	0.05
	0.05	0	0	0	0.95
T^2=	1	0	0	0	0
	0.17	0.16	0.61	0.06	0
	0.1128	0	0.6724	0.2088	0.006
	0.0601	0	0	0.8464	0.0935
	0.0975	0	0	0	0.9025
T^4=	1	0	0	0	0
	0.269614	0.0256	0.507764	0.187752	0.00927
	0.201781	0	0.452122	0.317125	0.028972
	0.120085	0	0	0.716393	0.163522
	0.185494	0	0	0	0.814506
T^5=	1	0	0	0	0
	0.308736	0.01024	0.429166	0.233664	0.018194
	0.23987	0	0.37074	0.34601	0.04338
	0.149753	0	0	0.659082	0.191166
	0.226219	0	0	0	0.773781
T^10=	1	0	0	0	0
	0.453949	0.000104858	0.163504	0.304892	0.07755
	0.390429	0	0.137448	0.356329	0.115794
	0.291697	0	0	0.434388	0.273914
	0.401263	0	0	0	0.598737
T^20=	1	0	0	0	0
	0.637888	1.09951E-08	0.02249	0.190735	0.148887
	0.594497	0	0.018892	0.203762	0.18285
	0.528319	0	0	0.188693	0.282988
	0.641514	0	0	0	0.358486

The formula for cell B7 has been highlighted, and it would be important to recall that using the MMULT() function requires the resulting matrix region to be highlighted, enter the mentioned formula and, press the Ctrl key and the Shift key simultaneously, then press the Enter key.

From these, the required state matrices can be obtained through further matrix multiplication as

S(0)=	0	1	0	0	0
S(5)=	0.308736	0.01024	0.4291665	0.233664	0.0181941
S(10)=	0.453949	0.00010486	0.1635038	0.304892	0.0775502
S(20)=	0.637888	1.0995E-08	0.0224904	0.190735	0.1488873

The probability that you will be an intern after 10 years, equals approximately 0.000105, from cell J17 above in S(10)

The probability that you will be a project manger in 5 years, equals approximately 0.018194, from cell M15 above in S(5)

The probability that you will be fired within 20 years, equals 1 minus the probability that you are in either of the states 1,2,3 or 4 after 20 years. That is because if you are in any of these states, it necessarily implies that you didn;t get fired in any of the years before. Hence using values in S(20), this equals