Respuesta :
Answer:
The probability that exactly four road construction projects are currently taking place in this city is 0.168.
Step-by-step explanation:
Let X denote the number of road construction projects that take place at any one time in a certain city.
It is provided that X follows a Poisson distribution with parameter λ = 3.
Then the probability mass function of X is:
[tex]p_{X}(x)=\frac{e^{-\lambda}\cdot \lambda^{x}}{x!};x=0,1,2,3...[/tex]
Compute the probability that exactly four road construction projects are currently taking place in this city as follows:
[tex]P(X=4)=\frac{e^{-3}\cdot 3^{4}}{4!}[/tex]
[tex]=\frac{0.04979\times 81}{24}\\\\=0.16804125\\\\\approx 0.168[/tex]
Thus, the probability that exactly four road construction projects are currently taking place in this city is 0.168.
Using the Poisson distribution, it is found that there is a 0.1954 = 19.54% probability that exactly four road construction projects are currently taking place in this city.
We have only the mean for the discrete distribution, hence, the Poisson distribution is used.
Poisson distribution
- In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by:
[tex]P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}[/tex]
The parameters are:
- x is the number of successes
- e = 2.71828 is the Euler number
- [tex]\mu[/tex] is the mean in the given interval.
In this problem, the mean is of 3, hence [tex]\mu = 3[/tex].
The probability that exactly four road construction projects are currently taking place in this city is P(X = 4), hence:
[tex]P(X = 4) = \frac{e^{-3}3^{4}}{(4)!} = 0.1954[/tex]
0.1954 = 19.54% probability that exactly four road construction projects are currently taking place in this city.
You can learn more about the Poisson distribution at https://brainly.com/question/13971530
