Respuesta :
Answer:
15.6%
Step-by-step explanation:
Since each day there is a 6% chance that Lisa smiles at him then that means that each day there is a 94% chance that Lisa does not smile at him. To find the probability of Milhouse going longer than a month (30 days) without a smile from Lisa we need to multiply this percentage in decimal form for every day of the month. This can be solved easily by putting 94% to the 30th power which would be the same, but first, we need to turn it into a decimal...
94% / 100 = 0.94
[tex]0.94^{30}[/tex] = 0.156
Now we can turn this decimal into a percentage by multiplying by 100
0.156 * 100 = 15.6%
Finally, we can see that the probability that Milhouse goes longer than a month without a smile from Lisa is 15.6%
Using the binomial distribution, it is found that there is a 0.1563 = 15.63% probability that Milhouse goes longer than a month without a smile from Lisa.
For each day, there are only two possible outcomes, either Lisa smiles at him, or she does not. The probability that she smiles at him on day is independent of any other day, hence, the binomial distribution is used to solve this question.
Binomial probability distribution
[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:
- On any given day independently there is a 6% chance that Lisa smiles at him, hence, [tex]p = 0.06[/tex].
- Considering a 30 day month, we have that [tex]n = 30[/tex].
The probability that Milhouse goes longer than a month without a smile from Lisa is the probability of no smiles during 30 days, that is, P(X = 0), hence:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{30,0}.(0.06)^{0}.(0.94)^{30} = 0.1563[/tex]
0.1563 = 15.63% probability that Milhouse goes longer than a month without a smile from Lisa.
To learn more about the binomial distribution, you can take a look at https://brainly.com/question/24863377
