in a recent report, joes, a memphis - style barbecue chain, states that 11% of its customers order for…

Explanation:

Step1: Identify the binomial - distribution parameters

The binomial - distribution formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. Here, $n = 6$, $p=0.11$, and $1 - p = 0.89$. We want to find $P(X\lt3)=P(X = 0)+P(X = 1)+P(X = 2)$.

Step2: Calculate $P(X = 0)$

$C(6,0)=\frac{6!}{0!(6 - 0)!}=1$. Then $P(X = 0)=C(6,0)\times(0.11)^{0}\times(0.89)^{6}=1\times1\times(0.89)^{6}\approx0.5025$.

Step3: Calculate $P(X = 1)$

$C(6,1)=\frac{6!}{1!(6 - 1)!}=\frac{6!}{1!5!}=6$. Then $P(X = 1)=C(6,1)\times(0.11)^{1}\times(0.89)^{5}=6\times0.11\times(0.89)^{5}\approx0.3387$.

Step4: Calculate $P(X = 2)$

$C(6,2)=\frac{6!}{2!(6 - 2)!}=\frac{6\times5}{2\times1}=15$. Then $P(X = 2)=C(6,2)\times(0.11)^{2}\times(0.89)^{4}=15\times0.0121\times(0.89)^{4}\approx0.1287$.

Step5: Calculate $P(X\lt3)$

$P(X\lt3)=P(X = 0)+P(X = 1)+P(X = 2)\approx0.5025 + 0.3387+0.1287=0.970$.

Answer:

$0.970$