Answer to Question #197886 in Statistics and Probability for krisha Billimoria

Question

A study found that 28% of car owners in Fiji had their cars washed professionally rather than do it themselves. If 18 car owners are randomly selected, find the probability that at most two people have their cars washed professionally.

Answer 1

Let $X=$ the number of car owners who have their cars washed professionally: $X\sim Bin(n, p).$

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

Given $p=0.28, n=18$

$$P(X\leq2)=P(X=0)+P(X=1)+P(X=2)$$

$$=\dbinom{18}{0}0.28^0(1-0.28)^{18-0}$$

$$+\dbinom{18}{1}0.28^1(1-0.28)^{18-1}$$

$$+\dbinom{18}{2}0.28^2(1-0.28)^{18-2}$$

$$=0.72^{18}+5.04\cdot0.72^{17}+11.9952\cdot0.72^{16}$$

$$\approx0.084195$$