in a recent report, joes, a memphis - style barbecue chain, states that 12% of its customers order for…

Explanation:

Step1: Identify the distribution

This is a binomial - distribution problem. Let (n = 6) (number of trials/sample size), (p=0.12) (probability of success, i.e., a customer orders for delivery), and (q = 1 - p=1 - 0.12 = 0.88). The binomial probability formula is (P(X = k)=C(n,k)\times p^{k}\times q^{n - k}), where (C(n,k)=\frac{n!}{k!(n - k)!}).

Step2: Calculate (P(X\leq2))

We need to find (P(X>2)=1 - P(X\leq2)). First, calculate (P(X = 0)), (P(X = 1)), and (P(X = 2)) and then sum them up. For (k = 0): [ \begin{align*} C(6,0)&=\frac{6!}{0!(6 - 0)!}=1\ P(X = 0)&=C(6,0)\times(0.12)^{0}\times(0.88)^{6}\ &=1\times1\times(0.88)^{6}\ &\approx0.464 \end{align*} ] For (k = 1): [ \begin{align*} C(6,1)&=\frac{6!}{1!(6 - 1)!}=\frac{6!}{1!5!}=6\ P(X = 1)&=C(6,1)\times(0.12)^{1}\times(0.88)^{5}\ &=6\times0.12\times(0.88)^{5}\ &\approx0.363 \end{align*} ] For (k = 2): [ \begin{align*} C(6,2)&=\frac{6!}{2!(6 - 2)!}=\frac{6\times5}{2\times1}=15\ P(X = 2)&=C(6,2)\times(0.12)^{2}\times(0.88)^{4}\ &=15\times0.0144\times(0.88)^{4}\ &\approx0.142 \end{align*} ] Then (P(X\leq2)=P(X = 0)+P(X = 1)+P(X = 2)\approx0.464 + 0.363+0.142 = 0.969).

Step3: Calculate (P(X>2))

[P(X>2)=1 - P(X\leq2)=1 - 0.969 = 0.031]

Answer:

(0.031)