1. What is a Markov Chain?

A Markov chain is a mathematical system that undergoes transitions between different states according to certain probabilistic rules. The defining characteristic is that the probability of transitioning to any particular state depends solely on the current state and time elapsed, not on the sequence of events that preceded it.

2. How Does the Calculator Work?

The calculator uses the Markov chain formula:

Where:

• \( P_0 \) — Initial probability vector
• Transition Matrix — Square matrix of transition probabilities
• \( n \) — Number of steps
• \( P_n \) — Probability distribution after n steps

Explanation: The calculator raises the transition matrix to the nth power and multiplies it by the initial probability vector to find the state probabilities after n steps.

3. Understanding Transition Matrices

Details: A transition matrix is a square matrix where each entry represents the probability of moving from one state to another. Each row must sum to 1, representing all possible transitions from that state.

4. Using the Calculator

Tips:

• Enter initial probabilities as comma-separated values (must sum to 1)
• Enter the transition matrix with one row per line, values comma-separated
• Matrix must be square and each row must sum to 1
• Number of steps must be a positive integer

5. Frequently Asked Questions (FAQ)

Q1: What are some applications of Markov chains?
A: Used in physics, chemistry, economics, game theory, genetics, weather forecasting, and many other fields to model random processes.

Q2: What's the difference between discrete and continuous Markov chains?
A: This calculator handles discrete-time Markov chains where transitions happen at fixed intervals. Continuous-time chains use different mathematics.

Q3: What is an absorbing Markov chain?
A: A chain where certain states cannot be left once reached. These require special analysis techniques.

Q4: How do I know if my matrix is valid?
A: It must be square, all entries between 0 and 1, and each row must sum exactly to 1.

Q5: What's the maximum size matrix this can handle?
A: The calculator can handle reasonably sized matrices, but very large matrices may cause performance issues.