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:<math>
\begin{align}
\mathbb{P}(x \mbox{ received} \mid y \mbox{ sent}) & {} = \frac{ \mathbb{P}(x \mbox{ received} , y \mbox{ sent}) }{\mathbb{P}(y \mbox{ sent} )} \\ \\
& {} = \frac{ \mathbb{P}(x \mbox{ received} , y \mbox{ sent}) }{\mathbb{P}(y \mbox{ sent})} \frac{\mathbb{P}(y \mbox{ sent})}{\mathbb{P}(x \mbox{ sent})} \\ \\
& {} = \frac{ \mathbb{P}(x \mbox{ received} , y \mbox{ sent}) }{\mathbb{P}(y \mbox{ sent})} \\ \\
& {} = \mathbb{P}(y \mbox{ received} \mid x \mbox{ sent}).
\end{align}
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