# 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.

Hence:

 $\ds p_3$ $=$ $\ds \dfrac {!3} {3!}$ $\ds$ $=$ $\ds \dfrac {\ds 3! \sum_{k \mathop = 0}^3 \dfrac {\paren {-1}^k } {k!} } {3!}$ Definition of Subfactorial $\ds$ $=$ $\ds \dfrac {3! \paren {1 - 1 + \dfrac 1 {2!} - \dfrac 1 {3!} } } {3! }$ $\ds$ $=$ $\ds \dfrac 2 6$ $\ds$ $=$ $\ds \dfrac 1 3$
$p_3$ is roughly $0.0345$ (or less than $\dfrac 1 {4!}$) away from the estimate of $\dfrac 1 e$.

$\blacksquare$