LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Two coins and a rule that picks tomorrow’s coin from today’s flip form a two-state Markov chain. A Markov chain is a process whose next state depends on the current state and not on the path that got there. This one is homogeneous: the one-step probabilities do not change with the date. The chance of using coin one on day two, starting from an equal mix on day one, is 0.65. The chance on a much later day is not a one-step number. It is an entry in a matrix power of the one-step table. Chapman-Kolmogorov is the reason the power is the right object, and induction is the reason that identification holds for every horizon, not just two steps.
Why it matters
Coin one lands heads with probability 0.7 and tails with probability 0.3. Coin two lands heads with probability 0.6 and tails with probability 0.4. On day one either coin is chosen with probability 1/2. The rule after that is deterministic given the flip: heads tomorrow means coin one, tails tomorrow means coin two. State 0 is coin one. State 1 is coin two.
The idea in plain words
The probability of state 0 on day two is the law of total probability. You were in state 0 on day one with probability 1/2 and then stayed, which contributes 0.7 times 1/2. Or you were in state 1 and moved to state 0, which contributes 0.6 times 1/2. The sum is 0.35 plus 0.30, which is 0.65. A four-leaf tree with the same weights gives the same 0.65. The one-step matrix was enough because day two is one step from day one.
Day 27 is 26 steps from day one. The one-step matrix does not contain a 26-step probability. The lecture defines a family of matrices whose entry in row i and column j is the probability of being in j after n steps, given that you started in i. The parentheses around n are part of the name. They do not yet mean “multiply the matrix by itself.” At this stage the entries are unknowns that the definition promises exist.
Chapman-Kolmogorov fills in the structure. The probability of going from i to j in n plus m steps equals the sum, over an intermediate state k, of the probability of going from i to k in m steps times the probability of going from k to j in n steps. The argument splits the clock into a stop at m and a later leg of length n. Conditional probability, conditioned further on the starting state, rewrites the joint event. The Markov property then drops the start. If you know you are in k at the intermediate time, the earlier state does not change the chance of the later leg. What remains is exactly the row-column sum that defines matrix multiplication. So the (n plus m)-step matrix equals the m-step matrix times the n-step matrix, as matrices, not entry by entry.
How to use it
Write the one-step matrix from the coins before you write any multi-day probability. Check that each row sums to 1. Then compute day two by hand, as 0.7 times 1/2 plus 0.6 times 1/2, and require 0.65. That is the unit test that the states are labeled the way you think.
What usually goes wrong
Raising 0.7 to the 26th power, or raising every entry to the 26th, is not the n-step matrix. The lecture spends the induction on that distinction. The parentheses start as a name. Only after Chapman-Kolmogorov and the induction do they become a matrix power.
Questions people ask
What is the probability of coin one on day two?
0.65. Half the time you start on coin one and stay with probability 0.7. Half the time you start on coin two and move with probability 0.6. The weighted sum is 0.65.
Why is the n-step matrix a matrix power?
Chapman-Kolmogorov says an (n plus m)-step transition is the matrix product of an m-step and an n-step transition. Induction extends the two-step case, which is the one-step matrix squared, to every n. The product is a matrix product.
What did the 26-step calculation give?
The lecture’s closed form, checked against a simulation, came out about 0.666 repeating for the probability of being on coin one. That uses the matrix power and the equal day-one mix. It is not the one-step 0.65 repeated by hand.