It isn't 2013!

Let $a_i$ denote the integer chosen from between 1 and i. And $S_{/a_i}$ denote $S - a_i$ .

We see that $a_{2014}$ is from the set $(1,2,3\ldots,2013,2014)$ which is equivalent to $(0,1,2,3, \ldots,2013) \pmod{2014}$ . So for each and every value of $S_{/a_{2014}}$ , there exists one and only one value of $a_{2014}$ for which $S = S_{/a_{2014}} + a_{2014} \equiv 0 \pmod{2014}$ . So the probability of S being divisible by $2014$ is $\frac{1}{2014}$ .

Likewise, we can conclude that there exist $2012$ values of $a_{2013}$ for which $S = S_{/a_{2013}} + a_{2013} \not \equiv 0 \pmod{2013}$ So the probability that S is not divisible by $2013$ is $\frac{2012}{2013}$

Therefore the probability that S is not divisible by 2013 but divisible by 2014 is $\frac{2012}{2013} \times \frac{1}{2014} = \frac{1006}{2027091}$

$a + b = 1006 + 2027091 = 2028097 \equiv \boxed{097} \pmod{1000}$