Question

40% 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 \(n = 8\) (number of trials), \(p=0.4\) (probability of success), \(C(n,k)=\frac{n!}{k!(n - k)!}\)

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

\(C(8,0)=\frac{8!}{0!(8-0)!}=1\)
\(P(X = 0)=1\times(0.4)^{0}\times(0.6)^{8}=0.6^{8}=0.01679616\)

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

\(C(8,1)=\frac{8!}{1!(8 - 1)!}=\frac{8!}{1!7!}=8\)
\(P(X = 1)=8\times(0.4)^{1}\times(0.6)^{7}=8\times0.4\times0.0279936 = 0.08957952\)

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

\(C(8,2)=\frac{8!}{2!(8-2)!}=\frac{8\times7}{2\times 1}=28\)
\(P(X = 2)=28\times(0.4)^{2}\times(0.6)^{6}=28\times0.16\times0.046656=0.20901888\)

Step5: Sum the probabilities

\(P(X<3)=P(X = 0)+P(X = 1)+P(X = 2)\)
\(P(X<3)=0.01679616+0.08957952 + 0.20901888=0.31539456\)

Answer:

\(0.315\)