Markov Chain N-Step Transition Calculator

How it is calculated

The matrix is read row by row as P = [[a, b], [c, d]]: row 1 holds the probabilities of going from state 1 to state 1 (a) and to state 2 (b), and row 2 the probabilities from state 2 (c to state 1, d to state 2). Every entry must be between 0 and 1 and each row must add up to 1, so a + b = 1 and c + d = 1, otherwise there is no result. The step count n must be a whole number of at least 1; 2.5 or 0 give no result. The tool raises P to the power n by repeated squaring to get Pn and prints it to four decimals, with entry (i, j) the probability of being in state j after n steps starting from state i. It also gives the long-run split π = [c / (b + c), b / (b + c)]; if b and c are both 0 it reports an identity chain with no mixing.

Common mistakes

• Entering a matrix whose columns add to 1 instead of its rows. The tool checks each row (a + b and c + d), so columns (0.9, 0.1) and (0.5, 0.5) typed as a = 0.9, b = 0.5, c = 0.1, d = 0.5 give row sums of 1.4 and 0.6 and no result.
• Reading entry (1, 2) as the probability of a step from state 2 to state 1. The first number is the row, the starting state, and the second the ending state.
• Confusing the n-step matrix with the steady state. After only 3 steps the rows are still different (0.8440 versus 0.7800), and they match the steady state only for large n.
• Using percentages such as 90 and 10. The entries must be probabilities between 0 and 1; a value outside that range gives no result.
• Assuming the steady state is reached from every chain. A chain that always flips, with a = 0 and d = 0, never settles, even though the formula still prints π = [0.5 0.5].