Hi, maths_buff.

Markov chains are not my specialty, I am hoping that Milos or one of the other members is better versed in these than I am.

Having said that, I think your probability matrix needs to be transposed:

            L    S
S = L [ 0.6   0.25 ]
      S [ 0.4   0.75 ]

So, if someone owns a large car at time 0 we will have

x(0) = [1]
          [0]

at time 1 (3 years hence):

x(1) =  [ 0.6   0.25 ] [1]  = [0.6]
           [ 0.4   0.75 ] [0]     [0.4]

at time 2 (6 years hence):

x(2) =  [ 0.6   0.25 ]^2  [1] =  [ 0.46   0.3375 ]  [1]  = [0.46]
            [ 0.4   0.75 ]      [0]     [ 0.54   0.6625 ]  [0]     [0.54]

(Which has the values you had already mentioned)

OK, well, we still have your "8 year" problem ...

... you could cheat and work out the probabilites at 6 and 9 and ratio in between.

Or, you could work out an equivalent matrix that works on 1 year intervals.

In other words, what is the Matrix "P" where:

P^3 = [0.6 0.25]
          [0.4 0.75]

(This is just an off-the-cuff idea, may not be rigorous)

Just for Fun, I worked out that matrix "P" where

P^3 = [0.6 0.25]
          [0.4 0.75]

It is (approximately):

[0.818    0.114]
[0.182    0.886]

So, what is P^6 ?

[0.460    0.338]
[0.540    0.662]

And P^8 is:

[0.422    0.362]
[0.578    0.638]

And P^9 is

[0.411    0.369]
[0.589    0.631]

There is some drift due to calculation accuracy, but if you worked P out more accurately you may have something workable. But  there may well be a rigorous way to do this rather than my "hey, lets use Excel and see what we get" approach

Here's another theory I have:

          Large  Small
Large [ 0.60   0.40 ]
Small [ 0.25   0.75 ]

There are two possible ways for a large car owner to own a large car in six years:

A) Someone has a large car now, buys a large car after three years, buys another large car after six years

B) Someone has a large car now, buys a small car after three years, buys another large car after six years

Therefore the probability is P(LL)P(LL) + P(LS)P(SL) = 23/50 like I suspected.

Maybe it's wrong; maybe it's right. ???

It matters and it doesn't matter; after six years it doesn't matter because on average everyone will have a new car. But it terms of after 8 years, that the questionable part. Perhaps 99% of the population buys their car in that year.

Remember that we're only worrying about current large car owners who'll have a large car in eight years time.

I will try and fiddle with it because as long as the elements in each row add to 1, I can indicate what I did is correct.

I'll wait and see what other forum members say.

Once again, MathsIsFun, thank you!

maths_buff/kris was the owner of rover and has now gone bust and i blame rod