Understanding Markov Chains — A Journey Through Probability and Prediction

Have you ever wondered how computers make predictions—like when your music app seems to know the next song to play, or how weather forecasts estimate tomorrow’s conditions based on today’s patterns? Behind these everyday tools lies a mathematical idea called a Markov Chain, named after the Russian mathematician Andrey Markov, who developed the concept in the early 1900s.

What Is a Markov Chain?

A Markov Chain models systems that evolve step-by-step over time where the next state depends only on the current state, not on the full history. This principle is called the Markov Property:

“The future is independent of the past, given the present.”

To specify a Markov Chain, you need:

  • A set of possible states (for example, Sunny or Rainy).
  • Transition probabilities that describe how likely it is to move from one state to another in one step.

Transition Matrix (2×2 Case)

We often collect the transition probabilities into a transition matrix. For two states A and B, the matrix is: 

| P(A→A)   P(A→B) |
| P(B→A)   P(B→B) |

Each row sums to 1 because, starting from a state, the probabilities of going somewhere next must add up to 1.

A Simple 2×2 Example: Weather

Suppose the weather can be either Sunny (S) or Rainy (R). Based on observations:

  • If today is Sunny, there is a 0.7 chance it will be Sunny tomorrow and a 0.3 chance it will be Rainy.
  • If today is Rainy, there is a 0.4 chance it will be Sunny tomorrow and a 0.6 chance it will be Rainy.

Write the transition matrix as: 

| 0.7   0.3 |
| 0.4   0.6 |

Row 1 corresponds to transitions from Sunny; Row 2 corresponds to transitions from Rainy. Note each row sums to 1 (0.7 + 0.3 = 1 and 0.4 + 0.6 = 1).

Predicting the Next Day

Let the current-day state vector be: 

[ p(Sunny)   p(Rainy) ]

To get tomorrow’s probabilities, multiply the current state vector by the transition matrix. For example, if today is certainly Sunny, the state vector is: 

[ 1   0 ]

Now multiply: 

[ 1  0 ] × | 0.7  0.3 |
            | 0.4  0.6 |
= [ 0.7  0.3 ]

Interpretation: Tomorrow there is a 0.7 chance of Sunny and a 0.3 chance of Rainy. Repeating this multiplication step models how probabilities evolve over multiple days—this sequence of steps is the “chain.”

Why Markov Chains Matter

Markov Chains offer a simple way to model uncertainty over time. They appear in fields such as weather prediction, queueing and reliability, finance, biology, speech recognition, and web search ranking. Though Andrey Markov worked long before modern computing, his ideas form a foundation for today’s data science and machine learning.

Final Thought

When you see a prediction—whether it is tomorrow’s weather or a recommended song—remember Markov’s elegant insight: “The future depends only on the present.” Knowing where you stand now often tells you a great deal about what comes next.

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