Transmission Line Information Loss
A binary input - binary output channel which flips bits with equal probabilities is known as a Binary Symmetric Channel.
 

(Image Credit: Wikipedia - Binary Symmetric Channel)

The channel capacity $C$ is the maximum (actually, a supremum) mutual information between the channel inputs and the channel outputs over input distributions. You can't get information through a channel more than its capacity. Intuitively, this says that by designing the way you put inputs into the channel well, you can maximize the amount of stuff coming out of the channel that you can recover (and this limit is the capacity). 

Mathematically, this is
$$C = \sup_{p(x)} I(X;Y)$$
where $p(x)$ is probability distributions on the input $X$ and $Y$ is the output of the channel. 

In this case, you can write the input of the channel as $P(X=0) = q$ and $P(X=1) = 1-q$. With this distribution, you can see that $P(Y=0) = P(X=0 \text{ and no crossover}) + P(X = 1 \text { and crossover}) = q(1-p) + (1-q)p$ and $P(Y=1) = 1-P(Y=0)$. The mutual information between $X$ and $Y$ is given by $I(X;Y)=\sum_{x,y} p(x,y) \log \frac{p(x,y)}{p(x)p(y)}$ where $p(x,y)$ is the joint distribution of $x$ and $y$, and $p(x)$, $p(y)$ are the distributions of $x$ and $y$ respectively. By writing down $I(X;Y)$ and then maximizing over $q$, you will see $C = 1 - H(p)$ where $H(p)$ is the binary entropy function $-p \log_2 p - (1-p) \log_2 (1-p)$. Then, in your case, $p = \frac{1}{2}$ where $H(p) = 1$, so the capacity is $0$. 

If you're interested in learning more about this, a few good books to look (at very different levels of difficulty):


Elements of Information Theory (Second Edition), by Tom Cover and Joy Thomas (Easy, Probably Best Starting Point)
Information Theory and Reliable Communication, by Robert Gallegar (A bit harder)
Information Theory: Coding Theorems for Discrete Memoryless Channels by Czisar and Korner (Pretty hard)
Information Theory by Robert B. Ash (Dover book, Easy)


And an unrelated book which I love, but couldn't keep myself from mentioning, is David Mackay's Information Theory, Inference and Learning Algorithms (free on his website), even though it doesn't go over channel capacity.