Question
An audit database holds 20 records, of which 8 have a red compliance flag. An auditor draws 7 records uniformly without replacement and seals them in a bundle. A second auditor then chooses 3 of those 7 uniformly at random to open. Exactly 2 of the 3 opened records are red.
The remaining 4 records in the bundle stay sealed. Conditional on what was revealed, what is the probability that at least two of those four sealed records are red? Show why the first draw of seven and the later random disclosure cannot be treated as seven independent draws with replacement.
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Hint 1
Once the three opened records are known to contain two red and one unflagged record, how many red and unflagged records remain among the seventeen unopened records in the original database?
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Solution
A uniform seven-record bundle followed by a uniform selection of three bundle members to open is equivalent to first choosing an ordered three-record opened subset and then choosing four more sealed records from the remaining population, uniformly without replacement. More formally, every pair of disjoint subsets of sizes three and four has the same probability under the two-stage sampling rule. Conditioning on the opened subset's colour count preserves exchangeability among the other seventeen records.
The three opened records contain two red and one non-red. Therefore the unobserved population now contains \(8-2=6\) red and \(12-1=11\) non-red records, a total of 17. If \(K\) is the red count among the four sealed records, then
\[\Pr(K=k\mid\text{opened: 2 red})=\frac{\binom6k\binom{11}{4-k}}{\binom{17}4},\quad 0\le k\le4.\]
For the requested event, the numerator is
\[\binom62\binom{11}2+\binom63\binom{11}1+\binom64\binom{11}0=15(55)+20(11)+15=1060.\]
There are \(\binom{17}4=2380\) equally likely sealed subsets. Hence
\[\Pr(K\ge2\mid\text{opened: 2 red})=\frac{1060}{2380}=\boxed{\frac{53}{119}}.\]
The complementary check counts zero red, \(\binom{11}4=330\), and one red, \(\binom61\binom{11}3=6(165)=990\); \(2380-(330+990)=1060\). Independence would be false: after opening a red record, the chance that another of the finite remaining records is red changes. The event concerns the four sealed bundle records, not all seventeen unopened database records.
Assumptions
The first seven-record bundle is uniformly sampled without replacement from the known population.
The three disclosure records are uniformly sampled from that bundle independently of their unseen flags.
Only the red-count (2) among the three opened records is conditioned on; no selection or disclosure depends on flags.
Final Answer
\(\frac{53}{119}\), approximately 44.54%.