# 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 $$A+B+C+D+E$$.

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