Answer to Question #200550 in Statistics and Probability for Matthew Kayton

Question

It has been found that 1 out of every 20 people who visit a retail website make a purchase worth R500 or more. If we randomly select a sample of 15 visitors to the website, what is the probability that no more than three of the visitors will make a purchase worth R500 or more? Interpret your answer.

Answer 1

Let $X=$ the number of the visitors who will make a purchase worth R500 or more: $X\sim Bin(n,p)$

Given $p=1/20=0.05, n=15.$

$$P(X=x)=\dbinom{n}{x}p^x(1-p)^{n-x}$$

$$P(X\leq 3)=P(X=0)+P(X=1)$$

$$+P(X=2)+P(X=3)$$

$$\dbinom{15}{0}0.05^0(1-0.05)^{15-0}$$

$$+\dbinom{15}{1}0.05^1(15-0.05)^{15-1}$$

$$+\dbinom{15}{2}0.05^2(1-0.05)^{15-2}$$

$$+\dbinom{15}{3}0.05^3(15-0.05)^{15-3}$$

$$=0.46329123016+0.36575623434$$

$$+0.13475229686+0.03073297999$$

$$\approx0.994533$$