AHL 4.19—Transition matrices – Markov chains

Row 1 is 'from X', row 2 is 'from Y', in the order (X, Y) for both rows and columns.

25% of X moves to Y, so 75% stays at X: row 1 is $(0.75,0.25)$.

40% of Y moves to X, so 60% stays at Y: row 2 is $(0.40,0.60)$.

$0.75+0.25=1$ and $0.40+0.60=1$, confirming a valid transition matrix.

This gives $\left(\begin{array}{cc}0.75& 0.25\\ 0.40& 0.60\end{array}\right)$, which is option A.

This swaps the 'stay' and 'leave' probabilities in each row, putting the leaving probability first instead of the staying probability.

This uses 0.60 instead of 0.40 in row Y, incorrectly copying the stay probability of Y into the wrong position.

This swaps the two rows entirely, describing 'from Y' behaviour in row 1 and 'from X' behaviour in row 2.

All incorrect options confuse which probability belongs to staying versus switching, or which row corresponds to which state.

In a valid transition matrix, every row must sum to exactly 1.

Row 1 is $(0.5,a)$, so $0.5+a=1$.

$a=1-0.5=0.5$.

Row 2 is $(0.3,0.7)$ which sums to 1, confirming consistency.

Therefore $a=0.5$, matching option A.

$a=0.3$ mistakenly copies the value from row 2 column 1 instead of using the row 1 sum condition.

$a=0.7$ mistakenly copies the value from row 2 column 2 rather than solving row 1's own equation.

$a=0.2$ would come from incorrectly assuming the column sums to 1 (column-stochastic convention) instead of the row.

These errors confuse row-based and column-based sum conditions or misread the matrix layout.