Topic 4.19 · teacher page · Higher Level

Three starts, one ending

Why steady state questions never give you a starting distribution, and the equation students always leave out.

The one thing to do with the animation

Ask which of the three districts ends up with the most BTS users.

Starting from all BTS, all driving, and half and half, students expect three different answers. All three lines converge on 0.6 within about seven weeks and then stay there.

That is the whole justification for solving Ts = s with no initial vector in sight. Run it before the algebra, not after, and the algebra arrives as an explanation of something they have already seen rather than a rule.

The equation they leave out

Ts = s gives two equations that say the same thing, so it cannot determine two unknowns. You must add a + b = 1, and it is a mark in its own right.

Without it they get a direction, not a distribution, and a scaled multiple of (0.6, 0.4) is not a set of probabilities. If a student's answer does not sum to 1, this is why.

The answers

1. Driving to BTSThe top of the second column, so 0.3.
2. One week from all driving0.8 × 0 + 0.3 × 1 = 0.3.
3. Long-run BTS proportionSolving 0.8a + 0.3(1 − a) = a gives a = 0.6.
4. No starting distribution givenC. The steady state does not depend on where the chain starts.

Where the marks go

1 markSetting up Ts = s.

1 markWriting a + b = 1 as well. Both are needed and both are credited.

1 markAnswering in context: “in the long run 60% use the BTS”, not “0.6”.

What each wrong answer tells you

They giveWhat it means
0.7 (Q1)Gave the stay-with-driving probability. They read down the column rather than across to the row asked for.
0.2 (Q1)The other direction, BTS to driving.
0.8 (Q2)Applied the stay-on-the-BTS probability to a district with nobody on the BTS.
0.6 (Q2)Gave the long-run answer to a one-week question. Worth pointing at the green line, which starts at 0 and reaches 0.3 after one step.
0.5 (Q3)Assumed symmetry. The matrix is not symmetric, and this is the guess to pre-empt.
A (Q4)Believes the question is incomplete. The three converging lines are the answer.
An answer not summing to 1They omitted a + b = 1. Diagnose it from the sum alone, in two seconds.

Other things they will say

"Which way round does the matrix go?" Check which direction sums to 1. Columns summing to 1 means the state is a column and T goes on the left; rows summing to 1 means the state is a row and T goes on the right. Both appear in textbooks, so the check matters more than the convention.

"Does every chain have a steady state?" No. An absorbing state swallows everything, and a chain that strictly alternates cycles for ever. Every examination question will be well behaved, but knowing the exceptions is what makes the convergence meaningful.

"Can I just use a high power of T?" Yes, and it is a good check: T²⁰ has columns that are all nearly the steady state. Do it once alongside the algebra so they see the two methods agree.

A possible order

 What is happening
1Predict which district wins. Run the three lines. Let the convergence be the surprise.
2Reading a transition matrix, with the which-way-round check.
3Powers of T for a specific week, on calculators, and the first two questions.
4Solving for the steady state, with a + b = 1 written every time.
5A high power of T as a check, and a three-state example if the group is quick.

Two things not to say

Do not solve Ts = s without the sum-to-1 equation on the board. Students copy what they see and then cannot finish.

Do not say the steady state is “where it ends up” without adding “whatever it started as”. The second half is the content.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

  1. Read the matrixOf BTS users 80% stay; of drivers 30% switch to the BTS. State the transition matrix with BTS first.
    Columns are the starting state: [[0.8, 0.3], [0.2, 0.7]].
  2. One stepA district starts all driving, so s₀ = (0, 1). Find the proportion on the BTS after one week.
    0.8(0) + 0.3(1) = 0.3

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

  1. Two stepsFor the same start, find the BTS proportion after two weeks.
    0.8(0.3) + 0.3(0.7) = 0.45, climbing towards the steady state.
  2. Steady stateSolve for the long-run BTS proportion.
    0.8a + 0.3(1 − a) = a gives 0.5a = 0.3, so a = 0.6 and the split settles at 60% BTS, 40% driving.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

  1. No starting vectorA question asks for the steady state and gives no initial distribution. Explain why that is not missing information.
    The steady state is a property of the matrix alone. Three districts starting all BTS, all driving and half and half all converge on 0.6, so the start affects how long it takes and not where it ends.
  2. The equation they leave outSolving Ts = s gives two equations that say the same thing. State what must be added, and why it is a mark.
    a + b = 1. Without it you have a direction rather than a distribution, and any multiple of (0.6, 0.4) satisfies the matrix equation. If an answer does not sum to 1, this is the omission.

Practicalities

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