A diploma is organised by the College of Hogwarts on two years: $year1$ and $year2$. Each year an exam is organized in order to go to the upper level or be graduatie. Student has the probability $p$ to pass this exam. Those who fail repeat a year but must pass this time.
A student current year is observed every year in september. How to modelize by a Markov chain the schooling of a student?
I thought that as far as a student can repeat only once a year and that there are 2 years in the diploma, if we call X the current year and $Y$ the previous year, it should be:
$$P=\begin{pmatrix} 1-p & p & 0 &\\ 0 & 1-p & p &\\ 0 & 0 & 1 &\\ \end{pmatrix}$$
Then I want to find the different possible states in 2011 of a student who registered in 2009 and their probability.
- Having failed twice
- Having failed the first time but not the second one
- Having passed the first time but failed the second
- Having passed both time.
As far as there is 2 years of difference I tried to build up the transitional matrice $P^2$:
$$P^2=\begin{pmatrix} (1-p)^2 & 2p(1-p) & p^2 &\\ 0 & (1-p)^2 & p(1-p)+p &\\ 0 & 0 & 1 &\\ \end{pmatrix}$$
but here I don't understand what does $P(Y_2=j|X_0=i)$ means anymore.
The probability to have failed twice is $P(Y_2=1|X_0=1)=(1-p)^2$ The probability for one to have failed one's first year is $P(Y_2=2|X_=1)=2(1-p)p$ but why is it $2*...$(two times(1-p))? It doesn't make logically sense to me. The probability for one to have failed one's second year is now: $P(Y_2=2|X_2=2)$, but I don't understand the result either...