The student Union sells two types of Cola: Pospa and Poka.Given a
person last purchased Poka, there is a 90% chance that their next
purchase will be a Poka Cola. However if a person purchased Popsa
Cola, the chance that their next purchase is a Popsa Cola is only 70%.

a)If a person last purchased a Poka Cola, what is the probability that
the same person two purchases from now will purchase a Popsa Cola?

b) What are the Conditions for the sale of Cola reaching a steady state?

c) What are the steady state probabilities for the sale of the two types of cola?

d)if a person last purchased Popsa Cola, How many purchases will that
person make on avarage before purchasing a Poka Cola ?

e)Discribe how is the transition probability matrix used? 


Please show your working out for each question and discribe your
answere as as fully as possible.

This problem could be solved by Markov Chain modelling techniques.

1)There is something called transition probability which is defined as
the probability of one event changing into another or remain the same.
In the given problem: If we define purchasing Poka cola as event A and
Popsa cola as B,      P(ab) would be defined as a transition prob.
from a to b. There are four transition probabilities p(ab), p(ba),
p(aa), and p(bb). This could be represented in matrix form as follows,

p(aa)  p(ab)

p(ba)  p(bb)

For the given problem: Transition matrix P is,

	poka	popsa
poka	0.9	0.1
popsa	0.3	0.7

To answer the first question, There is intial state of having
purchased Poka cola, which is represented by State matrix V(initial),

poka	popsa
  1       0

The State after tow purchases / transitions is 

V(initial)P2(read as P power 2) =   = [0.84	0.16]

So the probability of purchasing the popsa cola after two purchases is 0.16

2)the steady state conditions:
In V matrix,

v1+v2+v3+... = 1
[v1 v2 v3   ]P = [v1 v2 v3   ]

3)To find out the steady state prob, IF you keep squaring the
transition matrix till the elements in matrix does not change, then
you get the steady prob,

P1
0.900	0.100
0.300	0.700

P2	
0.840	0.160
0.480	0.520

P4	
0.782	0.218
0.653	0.347

P8	
0.754	0.246
0.737	0.263

P16	
0.750	0.250
0.750	0.250
	
0.750	0.250
0.750	0.250

So the steady-state prob. are 0.75 0.25

4)that is nothing but steady-state prob of P(bb) (how long will he/she
continue in purchasing popsa cola) = 0.25

5) Just look at the explanation for first four questions.

thanks maniindram-ga 

I really think that you have helped me a lot with this question and I
do think that the problem is defenetlly solved and I now understand
how to solve similiar problems.

again thanks a lot