Question
A museum cannot complete an insurance claim unless three original documents are found. From prior tracking, the probabilities that the invoice, the valuation and the shipping receipt are missing are 10%, 15% and 20%, respectively. The documents could have been lost together in one incident, so the museum has no information about dependence between these three missing-document events. What is a guaranteed upper bound on the probability that the claim file is incomplete because at least one of the three documents is missing? Can the actual probability be determined exactly from the three percentages? Give two coherent ways the losses could be related that demonstrate your answer.
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Hint 1
The file is incomplete if at least one missing-document event happens. No assumption makes those events independent.
Check your answer
Solution
Let \(A,B,C\) denote the missing invoice, valuation and receipt. An incomplete file is the union \(A\cup B\cup C\). The union bound gives
\[
P(A\cup B\cup C)\le\min\{1,P(A)+P(B)+P(C)\}
=\min\{1,0.10+0.15+0.20\}=\boxed{0.45}.
\]
This bound is valid regardless of whether a common accident affects several documents.
To show that the exact probability is not determined, consider a uniform random number \(U\) on \([0,1]\). One possible dependence model is \(A=\{U<0.10\}\), \(B=\{U<0.15\}\), \(C=\{U<0.20\}\). Here the losses are nested, giving \(P(A\cup B\cup C)=0.20\). Another model uses disjoint intervals: \(A=\{0\le U<0.10\}\), \(B=\{0.10\le U<0.25\}\), \(C=\{0.25\le U<0.45\}\). The three marginal probabilities are unchanged, but their union has probability 0.45.
In general the union must be at least \(\max(0.10,0.15,0.20)=0.20\), because it contains each event. Thus both extremes are feasible and only the range \(\boxed{20\%\le P(\text{incomplete})\le45\%}\) is justified by these marginals. Summing individual risks as though they were disjoint, or multiplying complements as though they were independent, would add assumptions not supplied.
Assumptions
Only the three marginal probabilities are known; there is no assumed joint probability model.
The file is incomplete whenever at least one essential document is missing.
Final Answer
At most 45% by the union bound. The exact probability is not identified: with the stated marginals it can be as low as 20% or as high as 45%.