40% of consumers believe that cash will be obsolete in the next 20 years. assume that 6 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.)

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Accepted Answer

# Explanation:
## Step1: Identify the binomial formula
The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n = 6$ (number of trials, i.e., consumers selected), $p=0.4$ (probability that a consumer believes cash will be obsolete), and we want $P(X<3)=P(X = 0)+P(X = 1)+P(X = 2)$. The combination formula $C(n,k)=\frac{n!}{k!(n - k)!}$.
## Step2: Calculate $P(X = 0)$
$$
\begin{align*}
C(6,0)&=\frac{6!}{0!(6-0)!}=1\
P(X = 0)&=C(6,0)\times(0.4)^{0}\times(1 - 0.4)^{6-0}\
&=1\times1\times0.6^{6}\
&=0.046656
\end{align*}
$$
## Step3: Calculate $P(X = 1)$
$$
\begin{align*}
C(6,1)&=\frac{6!}{1!(6 - 1)!}=\frac{6!}{1!5!}=6\
P(X = 1)&=C(6,1)\times(0.4)^{1}\times(0.6)^{5}\
&=6\times0.4\times0.07776\
&=0.186624
\end{align*}
$$
## Step4: Calculate $P(X = 2)$
$$
\begin{align*}
C(6,2)&=\frac{6!}{2!(6-2)!}=\frac{6\times5}{2\times 1}=15\
P(X = 2)&=C(6,2)\times(0.4)^{2}\times(0.6)^{4}\
&=15\times0.16\times0.1296\
&=0.31104
\end{align*}
$$
## Step5: Sum up the probabilities
$$
\begin{align*}
P(X<3)&=P(X = 0)+P(X = 1)+P(X = 2)\
&=0.046656+0.186624 + 0.31104\
&=0.54432\approx0.544
\end{align*}
$$

# Answer:
$0.544$

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the binomial formula

Step2: Calculate \(P(X = 0)\)

Step3: Calculate \(P(X = 1)\)

Step4: Calculate \(P(X = 2)\)

Step5: Sum up the probabilities

Answer: