A dairy collects milk from villages P, Q and R in the ratio 5 : 3 : 2. The chances that a can from P, Q and R fails the quality test are 2%, 4% and 5%. Find the probability that a can chosen at random fails the test.

1. 1 · Write what is given and what is to be found. Let E₁, E₂, E₃: the can is from P, Q, R; F: the can fails. P(E₁) = 0.5, P(E₂) = 0.3, P(E₃) = 0.2. To find: P(F).
2. 2 · Draw the figure or sketch. Draw a tree: three branches marked 0.5, 0.3 and 0.2, each followed by a 'fails' branch marked 0.02, 0.04 and 0.05.
3. 3 · Write the theorem. Theorem of total probability: P(F) = Σ P(Eᵢ) P(F|Eᵢ).
4. 4 · Work out each branch.
   • P(E₁)P(F|E₁) = 0.5 × 0.02 = 0.010
   • P(E₂)P(F|E₂) = 0.3 × 0.04 = 0.012
   • P(E₃)P(F|E₃) = 0.2 × 0.05 = 0.010
5. 5 · Add the branches. P(F) = 0.010 + 0.012 + 0.010 = 0.032
6. 6 · Say what it means. About 32 cans in every 1000 would fail, which helps the dairy plan for spare stock.
7. 7 · Check and box the answer. The first branches add to 0.5 + 0.3 + 0.2 = 1. P(F) = 0.032.

Scoring tip: Turn the ratio 5 : 3 : 2 into probabilities 0.5, 0.3 and 0.2 first, and check that they add to 1.