How to play the following game? (placing balls into bins)
I think $x_1 = \dots = x_{\ell-1} = 0, x_{\ell} = y_1 = \dots = y_{\ell} = n$ is better.  You are guaranteed to choose red in all bins except for bin $\ell$, and bin $\ell$ has $n$ red balls and $n$ blue balls, so your overall odds of success are $1/2 &gt; 1/e$.