Question
A delivery service classifies parcels as easy (70% of all parcels) or difficult (30%). A dispatcher may send a parcel by urgent courier (U) or routine courier (R). Under the current policy, 20% of easy parcels and 80% of difficult parcels are sent urgently. Once the parcel type and courier choice are fixed, the probability of on-time delivery is:
| Parcel type | Urgent U | Routine R |
|---|---|---|
| Easy | 95% | 85% |
| Difficult | 60% | 30% |
Assume the table also describes what would happen if management changed the routing policy: for each type, forcing U instead of R does not change the corresponding success rate, and there are no interactions between parcels.
The manager observes that a parcel went out urgently under the current policy. What is its on-time probability? Compare this with the overall on-time probability under a new policy that sends every parcel urgently. Why are these not the same quantity, even though both refer to urgent deliveries?
Get hint
Hint 1
Urgent parcels are not a random cross-section of all parcels: difficult parcels are disproportionately routed urgently.
Check your answer
Solution
Let \(E\) and \(D\) denote easy and difficult parcels, and \(S\) mean on-time completion. Under the current policy,
\(\Pr(E\cap U)=0.7(0.2)=0.14\), \(\Pr(D\cap U)=0.3(0.8)=0.24\), so \(\Pr(U)=0.38\).
The joint probability of urgent and on time is \(0.14(0.95)+0.24(0.60)=0.133+0.144=0.277\). Conditional on observing an urgent dispatch, the success probability is consequently
\(\Pr(S\mid U)=0.277/0.38=\boxed{277/380\approx0.72895}\).
Under a policy that forces urgent delivery for every parcel, the type mix stays at 70% easy and 30% difficult. By the problem's explicitly stated policy-invariance rule, the success probability is
\(\Pr(S\mid \operatorname{force}U)=0.7(0.95)+0.3(0.60)=0.665+0.180=\boxed{0.845}\).
These expressions use different denominators and weights. Observed urgent parcels are only \(0.14/0.38=7/19\) easy, whereas the entire population is \(7/10\) easy. The currently urgent group is selected towards difficult cases; forcing the action changes that composition. For an additional consistency check, routine parcels have current success rate \([0.7(0.8)(0.85)+0.3(0.2)(0.30)]/0.62=0.494/0.62\approx0.7968\), even though urgent delivery improves the success rate within both types. This reversal is a selection effect, not evidence that urgent service is harmful. No unmeasured causal premise is needed because the problem supplies the policy-invariant type-specific rates explicitly.
Assumptions
Parcel type is determined before the route and is unaffected by any dispatch policy.
Route selection under the existing policy has the stated probabilities conditional on type.
The given on-time probabilities apply to both observed and externally forced routes within type, with no interference.
Final Answer
Currently observed urgent deliveries succeed with probability \(277/380\approx72.89\%\). Sending every parcel urgently yields \(169/200=84.5\%\) on-time delivery.