Transition Probability Matrix
However, using a tree diagram also has its limitations: if we want to calculate the probability after a month or even half a year, the tree diagram method will no longer be efficient. Therefore, mathematicians adopted the calculation method using Matrix. The matrix below is called the “transition probability matrix”.
$ \left(\begin{array}{cccc}P_{11}&P_{12}&...&P_{1n}\\P_{21}&P_{22}&...&P_{2n}\\...&...&...&...\\P_{m1}&P_{m2}&...&P_{mn}\end{array}\right) $
Just as its name implies, each element inside the transition probability matrix describes a transition probability from state to another. Here, $ P_{11} $represents the probability of event 1 occurring again on the second day after event 1 occurred on the first day; $ P_{21} $ represents the probability of event 1 occurring on the second day after event 2 occurred on the first day… and so on and so forth. Using this method, the transition probability matrix of the weather example can be written as:
