Question
Ava, Ben, Cleo, Dev and Eli are each assigned to exactly one of five numbered workstations, with every possible one-to-one assignment initially equally likely. The coordinator seals the assignment and later checks four statements against it. The reports are guaranteed correct:
- True: Ava was assigned station 2.
- False: Ben was assigned station 1.
- False: Cleo was assigned station 3.
- False: Dev was assigned station 1.
No reassignment occurs after the checks. Given these four reports, what is the probability that Eli is at station 1? Explain why multiplying four separate report probabilities would not give the right conditional probability.
Get hint
Hint 1
Fix Ava at station 2 and count permutations of the remaining four people among the remaining stations.
Check your answer
Solution
The initial \(5!\) bijections are equally likely. Conditioning on \(\text{Ava}=2\) leaves \(4!=24\) equally likely assignments of Ben, Cleo, Dev and Eli to stations \(\{1,3,4,5\}\).
Within these 24, define the forbidden events \(B_1\): \(\text{Ben}=1\), \(D_1\): \(\text{Dev}=1\), and \(C_3\): \(\text{Cleo}=3\). Each excludes \(3!=6\) assignments. \(B_1\cap D_1\) is impossible, since one workstation cannot contain two people. The intersections \(B_1\cap C_3\) and \(D_1\cap C_3\) each have \(2!=2\) assignments; all triple intersections are impossible. Hence inclusion–exclusion gives
\[
N_{\rm survive}=24-(6+6+6)+(2+2)=10.
\]
For the requested event \(\text{Eli}=1\), the forbidden \(\text{Ben}=1\) and \(\text{Dev}=1\) reports are automatic. Assign Ben, Cleo and Dev to stations 3,4,5: there are \(3!=6\) arrangements, of which \(2!=2\) place Cleo at the forbidden station 3. Thus \(N_{E_1}=6-2=4\), and
\[
\Pr(Eli=1\mid\text{reports})=\frac4{10}=\boxed{\frac25}.
\]
The reports are not independent observations of separate random assignments. For example, if Ben occupied station 1 then Dev could not, even before conditioning on the other reports. We condition on the intersection of reports by counting compatible complete bijections, not by multiplying marginal exclusion factors.
Assumptions
The original assignment is a uniformly random bijection, not five independent workstation choices.
The four report outcomes are exact observations of that one fixed assignment, with no reporter error.
Each numbered workstation holds exactly one volunteer and each volunteer occupies exactly one workstation.
Final Answer
\(2/5\). Exactly 10 one-to-one assignments survive the four reports; in 4 of them Eli occupies station 1.