Question
A library charges $3 for each complete day that a borrowed book remains overdue, but the total late fee is capped at $12. Each day after the due date, conditional on still having the book that morning, the borrower independently returns it sometime before closing with probability 1/4. A day on which the book is returned is not counted as a complete overdue day. The borrower keeps trying after the fee reaches its cap, and returns eventually with probability one.
What fee should the library expect to collect from this borrower? Give the exact amount as well as a value rounded to cents.
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Hint 1
If T is the day of return, counted from 1, the number of charged complete days is \(T-1\). The charge is \(3\cdot\operatorname{min}(T-1,4)\).
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Solution
Let \(T\in\{1,2,\ldots\}\) be the first post-due-date day on which the borrower returns the book. Conditional daily return probability \(p=1/4\) makes T geometric: \(P(T=t)=(3/4)^{t-1}(1/4)\). Because the return day is excluded, there are \(T-1\) chargeable complete days, but the fee stops at four days' worth. Define \(F=3\cdot\operatorname{min}(T-1,4)\) dollars.
For a nonnegative integer N, \(\operatorname{min}(N,4)=\sum_{k=1}^{4}1\{N\ge k\}\). Taking expectations gives the tail-sum identity
\[\begin{aligned}E[F]&=3\sum_{k=1}^{4}P(T-1\ge k)=3\sum_{k=1}^{4}(3/4)^{k}\\&=3[(3/4)(1-(3/4)^{4})/(1-3/4)]\\&=9(1-81/256)=1575/256\end{aligned}\] dollars ≈ $6.15.
For an independent direct distribution check, the possible fees are $0 for \(T=1\), $3 for \(T=2\), $6 for \(T=3\), $9 for \(T=4\) and $12 for \(T\ge5\). Thus
\[E[F]=3(3/16)+6(9/64)+9(27/256)+12(81/256)=1575/256\].
The cap limits charges, not the stochastic time until return. Counting the return day would shift the whole distribution and produce the wrong fee.
Assumptions
The daily conditional return chance remains 1/4 on every day until return, independent of earlier unsuccessful days.
The cap applies to collected dollars, not to the probability of return.
The return day itself never incurs a daily fee.
Final Answer
Expected fee = $\(\frac{1575}{256}\) ≈ $6.15 (more precisely $6.15234375).