Probability problem: Commission of $4$ members votes for a bill > There is a commission with $4$ members, namely $A, B, C, D$. This commission votes for a bill. Every member of the commission has $\dfrac{3}{4}$ probability of voting up and $\dfrac{1}{4}$ probability of voting down the bill. The bill will be applied only if it gets at least $3$ up votes. What is the probability of the bill to be apllied? My first idea is to do something like this: $$1 - (\text{probability of getting } 1 \text{ up vote}) - (\text{probability of getting } 2 \text{ up votes})$$ Any hint? Generally, is what I am thinking correct?

It is _not_ easier to use complementation in this instance.

If $X$ is the count of up-votes, then:

$$\begin{split}\mathsf P(X\geq 3)~ &=~1-\mathsf P(X=1)-\mathsf P(X=2)-\mathsf P(X=0)\quad&{\text{The correct way to use complements}\\\\\text{don't forget the case of zero up-votes}}\\\\[2ex]&=~\mathsf P(X=3)+\mathsf P(X=4)&\text{Anyway, its easier to just use this.}\end{split}$$