Question

39.7% of consumers believe that cash will be obsolete in the next 20 years. assume that 8 consumers are randomly selected. find the probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years. the probability is (round to three decimal places as needed.)

Explanation:

Step1: Identify the binomial probability formula

The binomial probability 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, $n = 8$, $p=0.397$, and $1 - p = 0.603$. We want to find $P(X\lt3)=P(X = 0)+P(X = 1)+P(X = 2)$.

Step2: Calculate $P(X = 0)$

$C(8,0)=\frac{8!}{0!(8 - 0)!}=1$.
$P(X = 0)=C(8,0)\times(0.397)^{0}\times(0.603)^{8}=1\times1\times(0.603)^{8}\approx0.017$.

Step3: Calculate $P(X = 1)$

$C(8,1)=\frac{8!}{1!(8 - 1)!}=\frac{8!}{1!7!}=8$.
$P(X = 1)=C(8,1)\times(0.397)^{1}\times(0.603)^{7}=8\times0.397\times(0.603)^{7}\approx0.090$.

Step4: Calculate $P(X = 2)$

$C(8,2)=\frac{8!}{2!(8 - 2)!}=\frac{8\times7}{2\times1}=28$.
$P(X = 2)=C(8,2)\times(0.397)^{2}\times(0.603)^{6}=28\times(0.397)^{2}\times(0.603)^{6}\approx0.212$.

Step5: Calculate $P(X\lt3)$

$P(X\lt3)=P(X = 0)+P(X = 1)+P(X = 2)\approx0.017 + 0.090+0.212 = 0.319$.

Answer:

$0.319$