The percentage of customers who respond to a company's messages is $20\%.$ In the past week, that company sent messages to $400$ customers. What is the probability that there are more than $50\%$ of those customers who responded the messages ?
It is not stated but we must assume that customer responses are independent (otherwise there is no solution). In that case, the probability is basically $\sum_{k= 201}^{400}\!\left ( 0.2 \right)^{k}\!\left ( 0.8 \right )^{400- k}\!\binom{400}{k}\!\cong 0.$ The intuition is that the distribution is very nearly normal with mean $400\cdot\frac{1}{5}= 80$ and standard deviation $\!\sqrt{400\!(1-\!\frac{1}{5})\!\frac{1}{5}}\!= 8,$ so having more than $200$ responses is a more than $15$ standard deviation event!
Is this right ??