Mathematics
Transition Probabilities
Quick fact
In a Markov chain, the transition probability from one state to another depends only on the current state, not on how you got there—this is called the memoryless property.
Why this is interesting
You've probably guessed tomorrow's weather based on today's conditions—that mental guess is essentially a transition probability.
Read the full explanation
Understanding Transition Probabilities
Imagine a frog hopping between three lily pads. Each time it jumps, there is a certain chance it lands on any pad, including staying put. These chances are transition probabilities. For each lily pad (state), the probabilities of where it goes next must sum to 1, because the frog must go somewhere. When you organize all these probabilities into a grid—each row showing the chances from a given starting state—you get a transition matrix. This matrix fully describes how the frog’s position evolves over time. Transition probabilities are simply the numbers in that matrix, and they capture the dynamics of any system that changes step by step with randomness.
A deeper explanation
Transition probabilities work because they encode the rule of system evolution under the Markov property: the future depends only on the present, not the past. This property makes modeling tractable—you don't need to remember history, only the current state and the probabilities of what comes next. These probabilities are not arbitrary; they often come from empirical data or physical laws. For example, in a random walk on a line, the transition probability to move left might be 0.5 and right 0.5. In a weather model, transition probabilities from 'sunny' to 'rainy' might be 0.3, based on historical records. Their importance lies in allowing us to compute long-term behavior, such as the probability of being in a certain state after many steps, or finding stable distributions. Without transition probabilities, we could not quantify or predict the unfolding of any stochastic process.