 will be close to their expectations with high probability. Then as <span class="math inline">\(N\)</span> becomes large enough, we will see that <span class="math inline">\(K_+\)</span> and <span class="math inline">\(K_-\)</span> become reasonably large.<br> First, by linearity of expectation we have <span class="math display">\[\begin{equation*}
\mathbb{E}[K_+] = \mathbb{E}[K_-] = \frac{N}{2^P}.
\end{equation*}\]</span> Next, by the union bound <span class="math display">\[\begin{eqnarray*}
\mathbb{P}(|K