Let $U_1,...,U_k$ be i.i.d uniform random variables over $\{0,1\}$. Let $W$ be an event that at least $k(\frac{1}{2} + \varepsilon)$ of these bits are one. Prove that $\Pr [U_j = 0 | W]\geq \frac{1}{2} + \varepsilon $ for every $j\in [k]$.
I tryed to use binomial coefficients but it did not help. Any seggestions? Thanks!
Since $U_i$'s are uniformly distributed, we can see that $\Pr[U_j=0]=\Pr[U_j=1]$.
Thus. $$\Pr[U_j=0\mid W]=\Pr[U_j=1\mid W]\geq\frac{k(1/2+\varepsilon)}{k}=\frac{1}{2}+\varepsilon. $$