A rail company's morning service is late the next day with probability \(0.35\) if it was late today, or with probability \(0.15\) if it was not late today....
Question 1 Report
A rail company's morning service is late the next day with probability \(0.35\) if it was late today, or with probability \(0.15\) if it was not late today. Monday's service is not late. The tree diagram covers Tuesday and Wednesday.
- Complete the missing probabilities on the tree diagram. (2)
- Work out the probability that the service is late on Wednesday. (3)
- Work out the probability that the service is late on at least one of Tuesday and Wednesday. (2)
- The company promises that the chance of being late on both days is below \(10\%\). Determine, with a reason, whether this promise holds. (2)
Answer Details
Since Monday's service was not late, the Tuesday branch uses the "not late today" rates, and each Wednesday branch depends on which outcome occurred on Tuesday; multiplying along a branch and adding branches that lead to the same event answers each part.
(a) Tuesday follows a not-late Monday, so its branches are \(P(\text{Tue late}) = 0.15\) and \(P(\text{Tue not late}) = 0.85\). From a late Tuesday, Wednesday's branches are \(0.35\) (late) and \(0.65\) (not); from a not-late Tuesday, they are \(0.15\) (late) and \(0.85\) (not), as completed on the tree above. [2 marks]
(b) Wednesday is late either via (Tue late, Wed late) or (Tue not late, Wed late): \(P(\text{Wed late}) = (0.15\times0.35)+(0.85\times0.15) = 0.0525+0.1275 = 0.18\). [3 marks]
(c) "At least one of the two days late" is the complement of "neither day late": \(P(\text{neither late}) = 0.85\times0.85 = 0.7225\), so \(P(\text{at least one late}) = 1-0.7225 = 0.2775\). [2 marks]
(d) \(P(\text{both late}) = 0.15\times0.35 = 0.0525 = 5.25\%\). Since \(5.25\% \lt 10\%\), the company's promise holds. [2 marks]
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