Question

What is the probability that at most two out of 10 customers will prefer the deluxe model

Answer 1

Let $X=$ number of customers who will prefer the deluxe model: $X\sim Bin (n, p).$

A marketing manager makes the statement that the long-run probability that a customer will the deluxe model to the standard model of product is 30%.

Given $n=10, p=0.3, q=1-p=1-0.3=0.7$

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

$$=\dbinom{10}{0}(0.3)^0(0.7)^{10-0}+\dbinom{10}{1}(0.3)^1(0.7)^{10-1}$$

$$+\dbinom{10}{2}(0.3)^2(0.7)^{10-2}$$

$$=(0.49+2.1+4.05)(0.7)^8\approx0.3828$$