Respuesta :
Using the binomial distribution, it is found that there is a:
a) 0.3439 = 34.39% probability he will receive at least one upgrade during the next two weeks.
b) 0.8671 = 86.71% probability that in a set of 20 flights, Sam will be upgraded 3 times or fewer.
For each flight, there are only two possible outcomes, either he receives an upgrade, or he dos not. The probability of receiving an upgrade in a flight is independent of any other flight, 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.
An airline claims that there is a 0.10 probability that a coach-class ticket holder who flies frequently will be upgraded to first class on any flight, hence [tex]p = 0.1[/tex].
Item a:
He takes 4 flights, hence [tex]n = 4[/tex].
The probability is:
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
In which:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{4,0}.(0.1)^{0}.(0.9)^{4} = 0.6561[/tex]
Then:
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.6561 = 0.3439[/tex]
0.3439 = 34.39% probability he will receive at least one upgrade during the next two weeks.
Item b:
20 flights, hence [tex]n = 20[/tex].
The probability is:
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)[/tex]
Then:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{20,0}.(0.1)^{0}.(0.9)^{20} = 0.1216[/tex]
[tex]P(X = 1) = C_{20,1}.(0.1)^{1}.(0.9)^{19} = 0.2702[/tex]
[tex]P(X = 2) = C_{20,2}.(0.1)^{2}.(0.9)^{18} = 0.2852[/tex]
[tex]P(X = 3) = C_{20,3}.(0.1)^{3}.(0.9)^{17} = 0.1901[/tex]
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.1216 + 0.2702 + 0.2852 + 0.1901 = 0.8671[/tex]
0.8671 = 86.71% probability that in a set of 20 flights, Sam will be upgraded 3 times or fewer.
To learn more about the binomial distribution, you can take a look at https://brainly.com/question/24863377
