ITPEC FE Subject B October 2024 Question 1
From the answer group below, select the correct combination of answers to be inserted into A and B in the program. Here, the array indexes start at 1.
Bicycle ridership in a city is studied. Examination of several years of data revealed that 30% of the people who regularly ride bicycles in a given year do not regularly ride bicycles in the subsequent year. Additionally, 2% of the people who do not regularly ride bicycles in that year begin to ride bicycles regularly in the subsequent year. If 5,000 people ride bicycles and 100,000 people do not ride bicycles in a given year, then the following program calculates the number of cyclists in the subsequent year (cycling) and that of people who do not ride bicycles in the subsequent year (noncycling):
Answer group
Explanation
A: P[1,1] × N[1] + P[1,2] × N[2], B: P[2,1] × N[1] + P[2,2] × N[2] — each group's size next year is its row of the transition matrix multiplied by this year's population vector.
The array literal fills P row by row: P[1,1] = pa = 0.7 (cyclists who keep cycling), P[1,2] = pb = 0.02 (non-cyclists who start), P[2,1] = pc = 0.3 (cyclists who stop) and P[2,2] = pd = 0.98 (non-cyclists who stay off bikes). N[1] holds the 5,000 cyclists and N[2] the 100,000 non-cyclists.
Row 1 collects everyone who cycles next year: 0.7 × 5000 + 0.02 × 100,000 = 3500 + 2000 = 5500.
Row 2 collects everyone who does not: 0.3 × 5000 + 0.98 × 100,000 = 1500 + 98,000 = 99,500. The two add up to 105,000, the population this year, as they must.
Answer (a)
Why not others:
- •(b) — A is right, but B gives 0.98 × 5000 + 0.3 × 100,000 = 34,900: it applies the stay-off rate to cyclists and the drop-out rate to non-cyclists, and the total no longer comes to 105,000
- •(c) — swaps the columns in both rows: A gives 0.02 × 5000 + 0.7 × 100,000 = 70,100, applying the start rate to cyclists and the keep rate to non-cyclists; B is the same wrong sum as in (b)
- •(d) — mixes entries from different rows: A gives 0.02 × 5000 + 0.98 × 100,000 = 98,100 and B gives 0.3 × 5000 + 0.02 × 100,000 = 3500
- •(e) — A gives 0.3 × 5000 + 0.02 × 100,000 = 3500, the people who change group, and B gives 0.7 × 5000 + 0.98 × 100,000 = 101,500, the people who stay; the total still matches, which makes it look plausible
- •(f) — the rows are swapped: A computes the non-cyclists (99,500) and B the cyclists (5,500)
Key rule: When a transition matrix has the destination state as its row and the source state as its column, each new count is its row times the old counts: the row index stays fixed while the column index and the N index move together.
Math & Data Science · AI-written, may contain errors
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