Respuesta :
Answer:
a) There is a 14.4% probability that the first two-door car is the third one in this model delivered to this dealer this week.
b) The expected number of cars in this model until the first two-door car is delivered in this week is 2.5.
Step-by-step explanation:
The negative binomial distribution allows us to find the number of failures before a success.
It has parameters n and p, in which n is the number of trials and p is the probability of a success.
The probability that it takes n trials for x sucesses is given by the following formula:
[tex]P = C_{n-1, x-1}*(p)^x*(1-p)^{n-x}[/tex]
In which C_{(n-1),(x-1)} is the number of different combinatios of x-1 objects from a set of n-1 elements, given by the following formula.
[tex]C_{n,x} = \frac{(n-1)!}{(x-1)!(n-x)!}[/tex]
a. Find the probability that the first two-door car is the third one in this model delivered to this dealer this week.
There is a 40% that a car is a two-door car, so [tex]p = 0.4[/tex]
We want it to take 3 trials for a sucess, so [tex]n = 3, x = 1[/tex].
[tex]P = C_{n-1, x-1}*(p)^x*(1-p)^{n-x}[/tex]
[tex]P = C_{2, 0}*(0.4)^1*(0.6)^{2} = 0.144[/tex]
There is a 14.4% probability that the first two-door car is the third one in this model delivered to this dealer this week.
b. Find the expected number of cars in this model until the first two-door car is delivered in this week.
The expected number of trials for r sucesses with p probability is given by the following formula.
[tex]\mu = \frac{r}{p}[/tex].
Here, we want the expected number of trials for 1 sucess, with 0.40 probability.
S
[tex]\mu = \frac{1}{0.4} = 2.5[/tex]
The expected number of cars in this model until the first two-door car is delivered in this week is 2.5.
