Question
A charity receives three independent all-or-nothing pledges. Donor A will give $40 with probability \(1/2\), donor B $70 with probability \(1/3\), and donor C $100 with probability \(1/4\). Otherwise each gives nothing; no donor makes a partial payment.
A sponsor has offered a particular matching arrangement. If total donations \(D\) are $100 or less, the sponsor pays nothing. If \(D\) exceeds $100, the sponsor pays one dollar for each dollar above $100, but no more than $80 in total. In symbols, the sponsor pays \(M(D)=\min(80,\max(0,D-100))\) dollars.
Before anyone confirms their pledge, what is the expected amount the sponsor will pay? Show the cases that actually trigger the match; the sponsor does not match the entire donation total once the threshold is crossed.
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Hint 1
Enumerate which of the three donors pay. The donor amounts differ, so the sponsor's payment depends on more than the number who participate.
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Solution
Let \(D\) be the sum of paid pledges. There are \(2^3=8\) subsets of participating donors; independence lets us multiply the corresponding payment and nonpayment probabilities. Every subset with at most one donor totals at most \(100\), so its match is zero. The four positive cases are:
| Donors who pay (exactly) | Total donations | Probability | Sponsor payment |
|---|---|---|---|
| A and B | $110 | \((1/2)(1/3)(3/4)=1/8\) | $10 |
| A and C | $140 | \((1/2)(2/3)(1/4)=1/12\) | $40 |
| B and C | $170 | \((1/2)(1/3)(1/4)=1/24\) | $70 |
| A, B and C | $210 | \((1/2)(1/3)(1/4)=1/24\) | $80 (cap) |
Consequently
\(\mathbb E[M]=10(1/8)+40(1/12)+70(1/24)+80(1/24)\)
\(=30/24+80/24+70/24+80/24=260/24=\boxed{65/6\text{ dollars}}\).
A separate check is that the probabilities of the four remaining zero-payment cases (nobody, only A, only B, only C) are \(1/4\), \(1/4\), \(1/8\) and \(1/12\). Those total \(17/24\), while the four positive case probabilities total \(7/24\); altogether the eight disjoint probabilities sum to one. The $100 donation by C alone produces zero matching payment because the offer is strictly above \(100\), and the $210 total yields only \(80\), not \(110\), because of the cap.
Assumptions
The three donor decisions are mutually independent Bernoulli outcomes with the stated probabilities and amounts.
The sponsor's threshold is strict and its payment is capped at $80, paid only on donations exceeding $100.
Final Answer
Expected sponsor contribution: \(65/6\approx\$10.83\).