Respuesta :
Using the binomial distribution, it is found that there is a 0.4113 = 41.13% probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.
For each consumer, there are only two possible outcomes, either they believe cash will be obsolete in the next 20 years, or they do not believe it. The belief of each customer is independent of any other customers, hence the binomial distribution is used to solve this question.
What is the binomial distribution formula?
The formula is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
- x is the number of successes.
- n is the number of trials.
- p is the probability of a success on a single trial.
Researching the problem on the internet, we have that:
- 40.4% of consumers believe that cash will be obsolete in the next 20 years, hence p = 0.404.
- 7 customers are surveyed, hence n = 7.
The probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years is given by:
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
In which:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{7,0}.(0.404)^{0}.(0.596)^{7} = 0.0267[/tex]
[tex]P(X = 1) = C_{7,1}.(0.404)^{1}.(0.596)^{6} = 0.1268[/tex]
[tex]P(X = 2) = C_{7,2}.(0.404)^{2}.(0.596)^{5} = 0.2578[/tex]
Then:
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0267 + 0.1268 + 0.2578 = 0.4113[/tex]
There is a 0.4113 = 41.13% probability that fewer than 3 of the selected consumers believe that cash will be obsolete in the next 20 years.
More can be learned about the binomial distribution at https://brainly.com/question/24863377
