40.3% of consumers believe that cash will be obsolete in the next 20 years. Assume that 7 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. Round your answer to three decimal places as needed.

Answer:

Step 1: Given Data

• Approximately 40.3% of consumers believe that cash will be obsolete in the next 20 years.
• The probability of success, $p=0.403$
• The sample size, $n=7$

Let the random variable $X$ be the number of consumers who believe that cash will be obsolete in the next 20 years. The formula for binomial probability is:

$P(x)=(\genfrac{}{}{0ex}{}{n}{x})p^x(1–p{)}^{n–x}$

Step 2: Calculating the Probability

(a) The probability that fewer than 3 of the selected consumers believe that cash will be obsolete can be calculated as:

$P(x<3)=P(x\le 2)$

Using the binomial probability formula:

$P(x<3)=\sum _{x=0}^2(\genfrac{}{}{0ex}{}{7}{x})(0.403{)}^x(1–0.403{)}^{7–x}$

Calculating the sum:

$P(x<3)\approx 0.413$

Using the Excel function =BINOM.DIST(2, 7, 0.403, 1), the probability is approximately 0.413.

Final Solution:

The probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years is $0.413$.