Answer to Question #339504 in Statistics and Probability for curties

Question

It has been reported that 70% of university students do volunteer work during their summer vacation. Four students are randomly selected to do volunteer work.

a. The probability that at least 1 student will do volunteer work this summer (correct to 3 decimal places) is 

b. The probability that exactly 3 graduates will not do any volunteer work this summer (correct to 4 decimal places) is 

c. The expected number of students (correct to 1 decimal place) who will not do volunteer work this summer is

Answer 1

Let $X=$ the number of students who do volunteer work during their summer vacation:$X\sim Bin(n, p).$

Given $n=4, p=0.7, q=1-p=0.3.$

a.

$$P(X\ge1)=1-P(X=0)$$

$$=1-\dbinom{4}{0}(0.7)^0(0.3)^{4-0}=0.992$$

b.

$$P(X^C=3)=P(X=1)=\dbinom{4}{1}(0.7)^1(0.3)^{4-1}$$

$$=0.0756$$

c.

$$E(X^C)=nq=4(0.3)=1.2$$