A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain.
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The probabilities of weather conditions, given the weather on the preceding day, can be represented by a transition matrix:
(P)_{i j} is the probability that, if a given day is of type i, it will be followed by a day of type j.
Notice that the columns of P sum to 1: this is because P is a stochastic matrix.
The weather on day 2 can be predicted in the same way:
In this example, predictions for the weather on more distant days are increasingly inaccurate and tend towards a steady state vector. This vector represents the probabilities of sunny and rainy weather on all days, and is independent of the initial weather.
The steady state vector is defined as:
Since the q is independent from initial conditions, it must be unchanged when transformed by P. This makes it an eigenvector (with eigenvalue 1), and means it can be derived from P. For the weather example: