Generally, symmmetry is presumed when using a two-sided confidence interval so that \(E_1 = E_2 = E\) and therefore the interval looks like
\begin{equation*}
P(\tilde{p} - E \lt p \lt \tilde{p} + E) = 1 - \alpha.
\end{equation*}
In this case, E is known as the margin of error.