Any hints re the question below would be greatly appreciated. I know we are dealing with $X\sim Bin(20, 0.4)$ and I know the "regular" formula $\binom{n}{x} \ p^x (1-p)^{n-x}$ to get a probability knowing $x$ but this seems to be reverse engineering the process and I can't figure out how to approach it.
Question
In a large restaurant an average of $2$ out of every $5$ customers ask for water with their meal. A random sample of $20$ independent customers is selected. Find the smallest value of $\alpha$ such that $P(X > \alpha) \ge 0.9$ where the random variable $X$ represents the number of these customers who ask for water.
The answer turns out to be trivial. The smallest value of $\alpha$ to satisfy the condition is 0.
\begin{align*} P(X>0) &= 1 - P(X \le 0) \\ &= 1 - P(X=0) \\ &= 0.99 \end{align*}
Interestingly, if the question were "Find the largest value of α such that.. " then there is no analytical solution but rather it has to be solved by computing all values of alpha until a probability of less than 0.9 appears. In that case the answer is the value of $\alpha$ just before that. If someone does know an analytical solution I'd be happy to hear about it.