Respuesta :
Answer:
a) 0.135 = 13.5% probability that during a given 1 min period, the first operator receives no requests.
b) 0.03185 = 3.185% probability that during a given 1 min period, exactly three of the six operators receive no requests
Step-by-step explanation:
To solve this question, we need to understand the Poisson distribution and the binomial distribution.
Poisson distribution:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given time interval.
Binomial distribution:
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
Poisson process with rate 2 per minute
This means that [tex]\mu = 2[/tex]
a. What is the probability that during a given 1 min period, the first operator receives no requests?
Single operator, so we use the Poisson distribution.
This is P(X = 0).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-2}*2^{0}}{(0)!} = 0.135[/tex]
0.135 = 13.5% probability that during a given 1 min period, the first operator receives no requests.
b. What is the probability that during a given 1 min period, exactly three of the six operators receive no requests?
6 operators, so we use the binomial distribution with [tex]n = 6[/tex]
Each operator has a 13.5% probability of receiving no requests during a minute, so [tex]p = 0.135[/tex]
This is P(X = 3).
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 3) = C_{6,3}.(0.135)^{3}.(0.865)^{3} = 0.03185[/tex]
0.03185 = 3.185% probability that during a given 1 min period, exactly three of the six operators receive no requests
