Question
Four volunteers are first assigned uniformly at random to four numbered stations: every one-to-one volunteer-to-station assignment has probability 1/24. Afterwards, the coordinator independently draws a reshuffle instruction. With probability 70%, stations 1 and 2 swap their assigned volunteers; with probability 20%, every volunteer moves one station forward in the cycle 1→2→3→4→1; and with probability 10%, the initial assignment is left unchanged.
Is the final complete assignment still uniform over all 24 possible assignments, despite the uneven reshuffle probabilities? Prove your answer. Would the same argument work if the starting assignment were fixed in advance?
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Hint 1
For any one of the three possible reshuffles, how many initial assignments map to a particular final assignment?
Check your answer
Solution
Let \(S\) denote one of the 24 complete one-to-one assignments, initially uniform, and let \(G\) be the independent random reshuffle. Each possible value \(g\) of \(G\) is a bijection on the set of assignments. Therefore, for any specified final assignment \(s\), the equation \(g(S)=s\) has exactly one solution \(S=g^{-1}(s)\). Conditional on any fixed reshuffle,
\[
P(g(S)=s\mid G=g)=P(S=g^{-1}(s))=\frac1{24}.
\]
Independence lets us use the same original uniform law in every condition. Averaging over the uneven reshuffle lottery gives
\[
P(G(S)=s)=0.7\frac1{24}+0.2\frac1{24}+0.1\frac1{24}
=\boxed{\frac1{24}}.
\]
Thus the probability distribution over final assignments is uniform; the lottery over permutations does not need to be uniform. The relevant property is that each allowed mapping is invertible and chosen without looking at the initial assignment.
If the starting assignment is fixed, this lottery can produce at most three final assignments, not all 24, so the final distribution is not uniform. Independence also matters: if the reshuffle were selected because of a particular initial assignment, conditioning on that reshuffle could distort the original uniform law. The preservation result applies to an independent mixture of bijections on an initially uniform state space.
Assumptions
The initial allocation is uniform among all \(4!\) one-to-one assignments.
The second-stage instruction is selected independently of the first-stage allocation.
Every allowed instruction is a permutation, and no volunteer is duplicated or removed.
Final Answer
Yes: every final one-to-one assignment still has probability \(1/24\). The starting uniformity and independent choice of a bijective reshuffle are essential; a fixed starting assignment would generally not become uniform.