Hi Jamie
In this example, we're trying to calculate the probability \( p_{0,2}^{(3)} \). In other words, the probability that an individual starting in State 0 ends up in State 2 after 3 steps.
One way we can calculate this is to consider the expected distribution of individuals who started in State 0 after 3 steps. So, because we're only considering individuals starting in State 0, the starting distribution across the states is (1,0,0), ie 100% in State 0.
When we take this distribution and multiply it by the transition matrix, we get the expected distribution across the states after one time step. This gives (1/4, 3/4, 0), which means of those starting in State 0, we expect 25% to state in State 0 after one year and 75% to move to State 1.
We repeat this process, taking this new distribution and multiplying by the transition matrix, which gives (1/4, 3/16, 9/16). This is now the expected distribution after 2 time steps. So after 2 years, we expect 25% to be State 0, 3/16 in State 1 and 9/16 in State 2.
Repeating one last time we get a distribution that looks like (*,*,9/16). This means that after 3 years we expect 9/16 of those individuals starting in State 0 to end up in State 2. This is exactly the probability we're looking for.
Changing the order that we perform the matrix calculations, this is actually the same as matrix multiplying the vector (1,0,0) by P^3, the transition matrix taken to the third power. This will just give the first row of the third power of the transition probability matrix, the last entry of which is our desired probability.
Hope this helps!
Andy
