What is the probability that the great-grandchild of middle class parents will be middle class? Markov chains can be used to answer these types of problems.

Features of a transition matrix It is a square, since all possible states must be used both as rows and as columns. All entries are between zero and one, inclusive, because all entries represent probabilities. The sum of the entries in any row must be one, since the numbers in the row give the probability of changing the state at the left to one of the states indicated across the top. Do assigned warm up problems

Transition diagram and matrix Transition diagram Transition matrix

Markov Chains A sequence of trials of an experiment is a Markov chain if 1. The outcome of each experiment is one of a set of discrete states; 2. The outcome of an experiment depends only on the present state, and not on any past states; 3. The transition probabilities remain constant from one transition to the next.

Transition probabilities gives the probabilities of a transition from one state to another in n repetitions of an experiment, provided the transition probabilities remain constant from one repetition to the next.

Probability vectors A probability vector is a matrix of only one row, having nonnegative entries, with the sum of the entries equal to one. Suppose a Markov chain has initial probability vector and transition matrix P. The probability vector after n repetitions of the experiment is

A problem using a probability vector If a student does homework one day, there is a 70% probability that he or she will do it again the next day. If a student does not do homework one day, there is a 60% probability that he or she will not do it again the next day. On Thursday, 75% of the students did their homework. What can you expect to happen on Friday?