In a city, 20% of office workers travel by metro, 50% by bus and 30% by two-wheeler. The probabilities of reaching office late are 0.1 for metro, 0.2 for bus and 0.3 for two-wheeler users. A worker reached office late. Find the probability that the worker travelled by metro.

1. 1 · Write what is given and what is to be found. Let E₁, E₂, E₃: travels by metro, bus, two-wheeler; A: is late. P(E₁) = 0.2, P(E₂) = 0.5, P(E₃) = 0.3; P(A|E₁) = 0.1, P(A|E₂) = 0.2, P(A|E₃) = 0.3. To find: P(E₁|A).
2. 2 · Draw the figure or sketch. Draw a tree with three branches (0.2, 0.5, 0.3), each followed by a 'late' branch (0.1, 0.2, 0.3).
3. 3 · Write the theorem. Bayes' theorem: P(E₁|A) = P(E₁)P(A|E₁) / Σ P(Eᵢ)P(A|Eᵢ).
4. 4 · Work out each branch.
   • 0.2 × 0.1 = 0.02
   • 0.5 × 0.2 = 0.10
   • 0.3 × 0.3 = 0.09
5. 5 · Find the total. P(A) = 0.02 + 0.10 + 0.09 = 0.21, by the theorem of total probability.
6. 6 · Find the posterior. P(E₁|A) = 0.02/0.21 = 2/21.
7. 7 · Say what it means. A late worker is most likely a bus user (10/21), so the office could look at bus timings first.
8. 8 · Check and box the answer. The posteriors 2/21 + 10/21 + 9/21 add up to 1. Answer: 2/21.

Scoring tip: Write the full formula with symbols before any numbers. The formula line alone earns a mark.