# Alphabet Soup

**Calculus**Level 5

\[\large P\left (\dfrac{1}{2}\right) - P(3) = \dfrac{A}{B} - C\ln3 + \dfrac{D}{E}\ln2\]

Let \(P(a), a \gt 0\) be the probability that for two real numbers \(x,y\) chosen uniformly at random from the interval \([-a,a]\) it is the case that \(xy \gt (x + y)\).

If the equation above holds true for positive integers \(A,B,C,D\) and \(E\) with \(\gcd(A,B) =\gcd(D,E) = 1\), find \(A+B+C+D+E\).