Respuesta :
Answer:
0.6676
Step-by-step explanation:
given that A string of Christmas lights con-
tains 20 lights. The lights are wired in series, so that
if any light fails, the whole string will go dark. Each
light has probability 0.98 of working for a 3-year
period.
Each light is independent of the other.
The probability that the string of lights will
remain bright for 3 years.
= Probability that each light does not fail
= multiplication (prob of one light does not fail)
=[tex]0.98^{20} \\=0.6676[/tex]
Using the binomial distribution, it is found that there is a 0.6676 = 66.76% probability that the string of lights will remain bright for 3 years.
For each light, there are only two possible outcomes, either it works or it does not work. The probability of a light working is independent of any other light, 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:
- There are 20 lights, hence n = 20.
- Each light has probability 0.98 of working for a 3-year period, hence p = 0.98.
The probability that all work is P(X = 20), hence:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 20) = C_{20,20}.(0.98)^{20}.(0.02)^{0} = 0.6676[/tex]
0.6676 = 66.76% probability that the string of lights will remain bright for 3 years.
More can be learned about the binomial distribution at https://brainly.com/question/14424710
