The fact that the question asks for an upper bound should immediately suggest an inequality is in order. Namely, in the case, we will use the Chebyshev inequality.
We will conduct 1000 polls, and our experiment will show 600 in favor of Obama and 400 in favor of McCain. How likely is this of happening?
Consider a random variable V, which is a poll of 1000 voters. That is, V is binomial with parameters n=1000, p=p<0.5. (Let the number of "heads" be the number of votes for Obama.) Our experiment turned up V = 600.
We also know that E(V) = np < 500. (Or, as later will be consequential: -E(V) > -500)
So, $ Pr(V = 600) = Pr(V - E(V) > 600 - 500) \leq Pr(|V - E(V)| > 100) \leq \frac{Var(V)}{100^2} $ (Chebyshev)
$ = \frac{1000 p (1-p)}{100^2} = 0.1 p (1-p) $
We say, "For any given p, our answer will be at most [some number]." The trick, then, is to find the maximum of our answer.
$ \frac{d}{dp} 0.1 p (1-p) = 0.1(1-2p) = 0 \Rightarrow p = 0.5 $
$ \frac{d^2}{dp^2} 0.1 p (1-p)|_{p=0.5} = -0.2|_{p=0.5} = -0.2 < 0 \Rightarrow $ p is a maximum.
So, $ \forall p, 0.1p(1-p) \leq 0.1(0.5)(1-(0.5)) = 0.025 $
Thus, $ Pr(V = 600) \leq 0.025 $.
