What does it mean to transmit reliably at rate $R$ through a binary symmetric channel $(\mathrm{BSC})$ with error probability $p$ ?

Assuming Shannon's second coding theorem (also known as Shannon's noisy coding theorem), compute the supremum of all possible reliable transmission rates of a BSC. Describe qualitatively the behaviour of the capacity as $p$ varies. Your answer should address the following cases,

(i) $p$ is small,

(ii) $p=1 / 2$,

(iii) $p>1 / 2$.