Question

a surgery has a success rate of 75%. suppose that the surgery is performed on four patients. what is the probability that the surgery is successful on exactly 2 patients?

Answer 1

Let $X=$ the number of patients who respond yes: $X\sim Bin(n, p).$

Given $n=4, p=0.75, q=1-p=0.25.$

$$P(X=2)=\dbinom{4}{2}(0.75)^2(0.25)^{4-2}=0.2109375$$

The probability that the surgery is successful on exactly 2 patients is $0.211.$