At a cafe, 60% of the customers order tea and the remaining 40% order coffee. A snack is also ordered by 20% of the tea customers and by 50% of the coffee customers. A customer chosen at random has ordered a snack. The probability that the customer ordered coffee is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{5}{8}\)
Let \(E_1\) and \(E_2\) be the events that the customer ordered tea and coffee, and let S be the event that a snack is ordered.
\(P(E_1)P(S|E_1) = 0.6\times 0.2 = 0.12\)
\(P(E_2)P(S|E_2) = 0.4\times 0.5 = 0.20\)
By Bayes’ theorem:
\( \displaystyle P(E_2|S) = \frac{0.20}{0.12+0.20} = \frac{0.20}{0.32} = \frac{5}{8} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{3}{8}\) — Incorrect. This is \(\frac{0.12}{0.32}\), the chance that the customer ordered tea.
Option (3) \(\displaystyle\frac{1}{2}\) — Incorrect. This is P(S | coffee), the condition taken the wrong way round.
Option (4) \(\displaystyle\frac{8}{25}\) — Incorrect. This is 0.32, the overall chance that a customer orders a snack.