Use a transition matrix to predict the next step practice questions | General Mathematics (Year 12, VCE Units 3 & 4, Transition matrices, Interpreting transition matrices)

Reading a transition matrix

A transition matrix records the proportion moving from each current state to each next state. Read a column for one source state, multiply by its current count and add all contributions that arrive at the required state

S_1=TS_0

Read source columns For an entry in row i and column j, column j is the current state and row i is the next state \(\displaystyle t_{21}=0.25\Rightarrow\text{25\% move from state 1 to state 2}\)
Convert percentages Write a percentage as a decimal proportion before multiplying \(\displaystyle 25\%=0.25\)
Find one contribution Multiply a transition proportion by the current count in its source state \(\displaystyle 0.25\times80=20\)
Add all arrivals Add the contributions from every current state that can reach the destination state \(\displaystyle 0.25\times80+0.80\times50=60\)

Worked example

A two-state population has 72 people in P and 45 people in Q. The transition matrix has columns for current states and rows for next states. Find the number in Q after one step

T=\begin{bmatrix}0.75&0.20\\0.25&0.80\end{bmatrix},\quad S_0=\begin{bmatrix}72\\45\end{bmatrix}

\(\displaystyle t_{21}=0.25\)The entry in row 2, column 1 gives the proportion moving from P to Q.
\(\displaystyle 0.25\times72=18\)The contribution from P to Q is 18 people
\(\displaystyle 0.80\times45=36\)The contribution staying in Q is 36 people
\(\displaystyle S_1(Q)=18+36=54\)Adding both arrivals gives 54 people in Q after one step

Common mistakes

Reversing row and column meanings

✕\(\displaystyle t_{12}=0.20\quad(P\to Q)\)✓\(\displaystyle t_{21}=0.25\quad(P\to Q)\)

Rows identify destination states and columns identify source states, so reversing them assigns each movement to the wrong transition entry

Counting only stayers

✕\(\displaystyle S_1(Q)=0.80\times50=40\)✓\(\displaystyle S_1(Q)=0.25\times80+0.80\times50=60\)

The next count in a state includes every contribution arriving there, including movement from other states

Writing a percentage as a whole number

✕\(\displaystyle 25\times80=2000\)✓\(\displaystyle 0.25\times80=20\)

A percentage is divided by 100 before it is used as a transition proportion

Difficulty levels

introRead a percentage transition and calculate one movement between two states

standardCombine a movement contribution and a stay contribution to find a next-state count

challengeCalculate all next-state totals in a three-state model and check conservation of the total

examChoose the transition interpretation or calculation that uses the stated matrix convention

Questions about this topic

Which direction does an entry in row i and column j represent?+

Column j is the current source state and row i is the next destination state, so the entry gives the proportion moving from j to i

Why are several products added for one next state?+

People in the next state can come from several current states, so each source contribution is calculated and then added

Why must the transition percentages in a column total 100%?+

A column describes all possible next states for one current state, so their proportions must account for the whole current group

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, General Mathematics, chapter 11: Transition matrices

In this chapter

11A Transition matrices and diagrams: Translate transition descriptions, diagrams and matrices 11A Transition matrices and diagrams: Check fixed-population and regular transition models 11B Interpreting transition matrices: Translate transition descriptions, diagrams and matrices Use a transition matrix to predict the next step 11C State recurrences and equilibrium: Check fixed-population and regular transition models Generate state matrices by recursion 11C State recurrences and equilibrium: Apply and reverse a state recurrence with external change Determine the long-run state of a transition model Construct a transition model with a specified equilibrium 11D Adjusted state recurrences: Apply and reverse a state recurrence with external change Construct and interpret Leslie matrices and life-cycle diagrams Generate and total Leslie population states Analyse long-run behaviour of a Leslie model