Get comprehensive solutions to your problems with IDNLearn.com. Join our Q&A platform to access reliable and detailed answers from experts in various fields.
Sagot :
Using the binomial distribution, it is found that there is a 0.9513 = 95.13% probability that less than three of these mortgages are delinquent.
For each mortgage, there are only two possible outcomes, either they are delinquent or they are not. The probability of a mortgage being delinquent is independent of any other mortgage, 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.
In this problem:
- 11% of mortgages were delinquent last year, hence p = 0.11.
- A random sample of eight mortgages was selected, hence n = 8.
The probability that less than three of these mortgages are delinquent is:
[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_{8,0}.(0.11)^{0}.(0.89)^{8} = 0.3937[/tex]
[tex]P(X = 1) = C_{8,1}.(0.11)^{1}.(0.89)^{7} = 0.3892[/tex]
[tex]P(X = 2) = C_{8,2}.(0.11)^{2}.(0.89)^{6} = 0.1684[/tex]
Then:
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.3937 + 0.3892 + 0.1684 = 0.9513[/tex]
0.9513 = 95.13% probability that less than three of these mortgages are delinquent.
You can learn more about the binomial distribution at https://brainly.com/question/24863377
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Thank you for visiting IDNLearn.com. We’re here to provide accurate and reliable answers, so visit us again soon.