Question
A portrait artist has three commissions. A small sketch takes 1 hour and pays $120, a charcoal portrait takes 2 hours and pays $180, and a full portrait takes 3 hours and pays $240. The artist starts at noon and attempts the three jobs in a uniformly random order, working continuously until closure.
The venue will unexpectedly close \(T\) hours after noon, where \(T\) is uniform on \([0,6]\) and independent of the job order. The artist keeps payment for a commission only if that particular portrait is fully finished by closing; an interrupted portrait earns nothing, and later jobs cannot be completed.
What is the expected total amount the artist receives?
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Hint 1
For any fixed order, a portrait is paid if the shared closing time exceeds its cumulative finish time. Take its payment times that survival probability.
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Solution
For a portrait finishing at time \(s\), the probability of payment is \(P(T\ge s)=1-s/6\). The single closing time applies to every portrait, so completion indicators are dependent, but linearity of expectation still lets us sum their expected payments.
The six orders and their conditional expected receipts (in dollars) are:
\[ \begin{array}{c|rrrrrr} \text{Order}&ABC&ACB&BAC&BCA&CAB&CBA\\\hline E[\text{receipts}\mid\text{order}]&190&180&180&160&160&150 \end{array} \]
For example, order ABC finishes jobs at hours 1, 3, and 6. Its expectation is \(120(5/6)+180(3/6)+240(0)=\$190\). Every order has probability \(1/6\), hence
\[ E[\text{receipts}]=\frac{190+180+180+160+160+150}{6}=\boxed{\$170}. \]
Equivalently, averaging finish-time survival probabilities across permutations gives completion probabilities \(5/12\) for A, \(1/3\) for B, and \(1/4\) for C. The independent check is \(120(5/12)+180(1/3)+240(1/4)=50+60+60=170\). Neither calculation assumes independent payment events.
Assumptions
All six orders are equally likely, independently of the single uniform closing time.
A portrait pays its full fee only if finished by closure; equality has probability zero.
Durations are deterministic and jobs cannot be resumed after closure.
Final Answer
The expected paid receipts are \(\$170\).