ITPEC FE Subject B October 2024 Question 1
ITPEC FE Subject B October 2024 — Question 1 of 20
Markov chain transition matrix: computing next-state populations by multiplying a state vector by a transition probability matrix.
The program models bicycle ridership transitions between two groups: cyclists (N[1] = 5000) and non-cyclists (N[2] = 100,000).
Transition probabilities:
- pc = 0.3 → probability a cyclist stops cycling
- pb = 0.02 → probability a non-cyclist starts cycling
- pa = 1 - pc = 0.7 → cyclist remains a cyclist
- pd = 1 - pb = 0.98 → non-cyclist remains a non-cyclist
Transition matrix P (row = destination, column = source):
| From cycling (col 1) | From noncycling (col 2) | |
|---|---|---|
| To cycling (row 1) | P[1,1] = pa = 0.7 | P[1,2] = pb = 0.02 |
| To noncycling (row 2) | P[2,1] = pc = 0.3 | P[2,2] = pd = 0.98 |
Next-year populations (matrix × vector):
- cycling = P[1,1] × N[1] + P[1,2] × N[2] = 0.7 × 5000 + 0.02 × 100000 = 3500 + 2000 = 5,500
- noncycling = P[2,1] × N[1] + P[2,2] × N[2] = 0.3 × 5000 + 0.98 × 100000 = 1500 + 98000 = 99,500
Why not others:
- (b) — B swaps P[2,2] and P[2,1], mixing up "stay non-cyclist" with "stop cycling"
- (c) — A uses P[1,2] × N[1] (applies "start cycling" rate to cyclists — wrong group)
- (d) — A uses P[2,2] (non-cyclist retention) in the cycling formula
- (e) — A starts with P[2,1] (dropout rate) as the first term for cycling count
- (f) — A uses P[2,1] and P[2,2] (both noncycling-row values) to compute cycling
Key rule: In a Markov transition matrix, next-state values are computed as row × column dot products: each row of P multiplied element-wise by the state vector N, then summed.
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