CT4 September 2014 Question 5i)

I am having trouble understanding some of the probabilities in the transition graph:

1P -> 1N = 0.7 (Finishing position is Other and it's coming from the state "promoted previous season" which is 0.6?)
2R -> 2N = 0.75 (I would of thought this probability is 0.5, regulated previous season -> other)
1N -> 1N = 0.85 (To me this should be 0.7, neither promoted/regulated previous season -> other)
2N -> 2N = 0.85 (To me this should be 0.7, neither promoted/regulated previous season -> other)

Thanks

 

Just realised the probabilities need to sum to 1 in the matrix and hence we get the above probabilities.

 

I’m not able to conceive the states at all. I have zero interest and knowledge about sports! Can someone explain the states

 

Hi, I too had some difficulty getting this, but it helped me to understand by breaking the transitions down in words:

Consider the following states:

- 1 N = In Division 1, Not Promoted previous season
- 1 P = In Division 1, Promoted previous season
- 2 R = In Division 2, Relegated previous season
- 2 N = In Division 2, Not Relegated the previous season

The screenshot attached shows the transition matrix relating to the above states.
Now breaking down some of the probabilities in the matrix.

Top Row

1N -> 2R: This transition is the team finishing Bottom in Division 1 (and so regulated to Division 2), coming from a state where the team was not promoted in the previous season. From the probability table in the question this is 0.15 (last column).

1N -> 1N: This transition is the team finishing either Top or any Other position than Bottom in Division 1 (hence remaining in Division 1), coming from a state where the team was not promoted in the previous season. The probabilities for this would be the sum of 0.15 + 0.7 = 0.85 (last column)

1N -> 1P: This transition is not possible, because for the team to get promoted they must be in division 2. So prob = 0.

1N -> 2N: This transition is not possible, as for the team not to be relegated in the previous season to be in division 2, the team must already be in division 2. Prob = 0.

For the other probabilities, similar logic follows. Hope this helps.