Hat-Check Problem/Examples/3

Example of Hat-Check Problem

$p_3 = \dfrac 1 3$

Proof

When $n = 3$, there are only three hats to hand back and $6 = 3!$ ways to return the hats to the owners.

$\begin{array}{c|ccc}

\ds Correctly Returned & Adam & Bill & Clyde \\ \hline

3 & A & B & C \\ \hline

1 & A & C & B \\ \hline

1 & C & B & A \\ \hline

1 & B & A & C \\ \hline

0 & B & C & A \\ \hline

0 & C & A & B \\ \hline

\end{array}$

From the table above, we see that there are only two permutations that irritate all parties involved.

Hence, $p_3 = \dfrac 2 6 = \dfrac 1 3$

$\blacksquare$