= Solution
A <reliable transmission rate> $R$ means that length-$n$ codes can carry $M_n$ messages with $n^{-1}\log_2M_n$ tending to at least $R$ while the <probability measure> of the decoding-error event tends to zero. By the <Shannon second coding theorem>, the supremum of reliable rates is the <binary symmetric channel capacity>
$$
C(p)=1-h_2(p),
\qquad
h_2(p)=-p\log_2p-(1-p)\log_2(1-p).
$$
For small positive $p$, the <binary entropy function> has the <asymptotic expansion>
$$
h_2(p)=p\log_2\frac1p+\frac{p}{\ln2}+O(p^2),
$$
so $C(p)$ starts at one bit per channel use and decreases rapidly from one as errors are introduced.
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