I was reading through the solution to a problem in a probability textbook and the author applied the following inequality without proving it: $$(1-\mathbb{P}(\text{Bin}(k,1/2)>s))^{t-1} \leq 2^{-|\Omega|(t\mathbb{P}(\text{Bin}(k,1/2)>s))}.$$
$\text{Bin}$ is the binomial distribution, and $s,t,k$ are natural numbers s.t. $t,k>0$, $0 < s < k$.
So the inequality above is equivalent to proving the following inequality, $$(t-1)\log_2(1-\mathbb{P}(\text{Bin}(k,1/2)>s))\geq -|\Omega|(t\mathbb{P}(\text{Bin}(k,1/2)>s)).$$ I am not really sure how to proceed here. Any help would be appreciated.