Question
Two club members each write a donation pledge on a sealed slip. Their amounts, A and B, are independent and uniformly distributed between $0 and $100. The organiser opens the slips and announces that A was the larger pledge, but reveals only that A's amount lies between $60 and $90.
Given both parts of that announcement, what is the probability that B, the smaller pledge, was more than $70? As a preliminary step, describe B's conditional distribution if the exact winning amount \(A=a\) were disclosed. Ignore ties, which have probability zero.
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Hint 1
Learning that A won changes the possible values of B: B must be below A. A winning amount near $60 cannot coexist with B above $70.
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Solution
Before the announcement, the joint density of \((A,B)\) is constant, \(1/10000\), on the $100-by-$100 square. Conditioning on the exact amount \(A=a\) and the event \(A>B\) restricts \(B\) uniformly to \([0,a]\). For \(0\le t<a\), this means \(P(B>t\mid A=a,A>B)=(a-t)/a\); for \(t\ge a\), it is zero.
With only the interval for A reported, use the joint selection region \(D=\{60\le A\le90,\;0\le B<A\}\). Because the joint density is constant, its probability is proportional to its area:
\[ \operatorname{Area}(D)=\int_{60}^{90} a\,da=\frac{90^2-60^2}{2}=2250. \]
The target region is \(N=\{70<B<A,\;60\le A\le90\}\). It is empty when \(A\le70\). Its area is
\[ \operatorname{Area}(N)=\int_{70}^{90}(a-70)\,da=\frac{20^2}{2}=200. \]
Therefore \(P(B>70\mid D)=200/2250=\boxed{4/45}\). This is not the unconditional chance that a uniform pledge exceeds $70 (30%): conditioning on having lost to A and on A's coarse winning band changes the relevant distribution. Nor can we treat A as uniformly distributed over its announced band after observing that A was the maximum; within that band, its conditional density is proportional to \(a\), since a larger A admits a larger set of losing B values.
Assumptions
The two pledge values are independent and continuous uniform random variables on \([0,100]\).
The announcement gives both the winner’s identity and the winning interval, and reveals no other information.
Ties and exact interval endpoints have probability zero.
Final Answer
For exact \(A=a\) with \(A>B\), \(B\sim\operatorname{Uniform}(0,a)\). With only \(60\le A\le90\) and \(A>B\) known, \(P(B>70)=4/45\approx8.89\%\).