Are you smarter than me?

@Mehul Chaturvedi : Please see point 2 of Suggestions for Sharers . Only mathematical expressions should be coded in LaTeX.

We know that $S_{n}=\dfrac{n*(n+1)}{2}.$ Now,we want that to be an even integer. $\Rightarrow\ (n)(n+1)$ should be divisible by $4.$ $\\\Rightarrow(n)(n+1)=4x.$ Let us look at the ways that can happen $:Case\ 1:n=4a$ this can happen when $a=1\rightarrow24.$ $Case\ 2:(n+1)=4a$ this can happen when $a=1\rightarrow25.$ You might be wondering why I didn't consider the case $n=2a,(n+1)=2b$ this is because $gcd(n,n+1)=1.$ Thus,the total number of required cases $=24+25=49$ and the total number of cases $=99\Rightarrow$ the required probability $=49/99.$

I have just written terms from S1 to S2 and observed a trend i.e. First 2 are odd next 2 are even , this means that this process will have equal number of even and odd numbers till each multiple of 4 thus until 96 , and further S97, S98 are odd and S99 in even . Thus total even no.s are 49 . Thus probability of even no is 49/99

So... were you not able to solve this yourself? If I were to get the same answer as you (the correct answer, obviously, since I am writing a solution), wouldn't that only mean that I am only (as far as we know) just as smart as you are?

Sn = n(n+1)/2 so, for an even Sn, n(n+1)/2 should be divisible by 2, n(n+1) should be divisible by 4, either n or n+1 should have 4 as a factor, counting all multiples of 4 and the preceding numbers (3,7,11 ...) we get 48. probability is 48/99