Finding a High Bound on Probability of Random Set first time user here.  English not my native language so I apologize in advance. Taking a final in a few weeks for a graph theory class and one of the sample problems is exactly the same as the $k$-edge problem.
We need prove that if each vertex selected with probability $\frac{1}{2}$, that the probability that except that we must find that the probability of the set being an
independent set is  $\geq (\frac{2}{3})^k$ (instead of $\frac{1}{2^k}$, like original problem).  
Here is what I am trying: I am looking into calculating the number of independent sets for each possible number of vertices $i$, for $i=0$ to $i=n$.  Once I calculate the probability of there being an independent set for the number of vertices $n$, I can take the expectation and union bound.

I was the one who asked the original question that you linked to. Just for fun I tried to see if I could prove a better bound.  I was able to prove a bound even better than that. Here's a hint: you can use indicator variables and the second moment method.
