Editing Binary symmetric channel (section)

== Definition ==
[[Image:Binary symmetric channel (en).svg|thumb|alt=Binary symmetric channel|upright=2|The binary symmetric channel sees each bit of a message transmitted correctly with probability 1–''p'' and incorrectly with probability ''p'', due to noise across the transmission medium.]]
A binary symmetric channel with crossover probability <math>p</math>, denoted by BSC<sub>p</sub>, is a channel with binary input and binary output and probability of error <math>p</math>. That is, if <math>X</math> is the transmitted [[random variable]] and <math>Y</math> the received variable, then the channel is characterized by the [[conditional probability|conditional probabilities]]:{{sfnp|MacKay|2003|p=4}}

:<math>\begin{align}
\operatorname {Pr} [ Y = 0 | X = 0 ] &= 1 - p \\
\operatorname {Pr} [ Y = 0 | X = 1 ] &= p \\
\operatorname {Pr} [ Y = 1 | X = 0 ] &= p \\
\operatorname {Pr} [ Y = 1 | X = 1 ] &= 1 - p
\end{align}</math>

It is assumed that <math>0 \le p \le 1/2</math>. If <math>p > 1/2</math>, then the receiver can swap the output (interpret 1 when it sees 0, and vice versa) and obtain an equivalent channel with crossover probability <math>1 - p \le 1/2</math>.