A courier company sends 70% of its parcels by road and 30% by rail. Of the parcels sent by road, 2% are delayed, and of those sent by rail, 6% are delayed. A parcel chosen at random is found to be delayed. The probability that it was sent by rail is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{9}{16}\)
Let \(E_1\) and \(E_2\) be the events that the parcel is sent by road and by rail, and let D be the event that it is delayed.
\(P(E_1)P(D|E_1) = 0.7\times 0.02 = 0.014\)
\(P(E_2)P(D|E_2) = 0.3\times 0.06 = 0.018\)
By Bayes’ theorem:
\( \displaystyle P(E_2|D) = \frac{0.018}{0.014+0.018} = \frac{0.018}{0.032} = \frac{18}{32} = \frac{9}{16} \)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{7}{16}\) — Incorrect. This is \(\frac{0.014}{0.032}\), the chance that the delayed parcel went by road.
Option (2) \(\displaystyle\frac{3}{10}\) — Incorrect. This is the share of parcels sent by rail, before the delay is known.
Option (4) \(\displaystyle\frac{4}{125}\) — Incorrect. This is 0.032, the overall chance that a parcel is delayed.