A vaccine, which is surrently in use, is known to be 80% effective. A pharmaceutical company has developed a new vaccine and the company claims 90% of the people given the new vaccine will develop the immunity. Suppose the claim of the company is true, and the newly developed vaccine is given to 20 people. What is the probability that at least 18 people will develop the immunity?
Progress: $n=20$, $p=.9$. $x=18,19,20$, $q=.1$ What is my next step in figuring this out? Do I have to do the $n!/(n-x)!p^xq^{n-x}$ for each value of $x$?
Yes. More correctly your model is a Binomial Distribution so:
$$\begin{align}\mathsf P(X=x) & = \dfrac{n!\,p^x\, q^{n-x} }{x!\, (n-x)!} \\[2ex] \mathsf P(X\geq 18) & = \dfrac{20!\,(0.9)^{18}\, (0.1)^{2} }{18!\, 2!} +\dfrac{20!\,(0.9)^{19}\, (0.1)^{1} }{19!\, 1!} +\dfrac{20!\,(0.9)^{20}\, (0.1)^{0} }{20!\, 0!}\end{align}$$
Simplify and evaluate.