Respuesta :
Answer:
a) There is an 81.87% probability that the instrument does not fail in an 8-hour shift.
b) There is a 45.12% probability of at least one failure in a 24-hour day.
Step-by-step explanation:
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.
a. What is the probability that the instrument does not fail in an 8-hour shift?
The mean for an hour is 0.025 failures.
For 8 hours, we have [tex]\mu = 8*0.025 = 0.2[/tex]
This probability is P(X = 0).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-0.2}*(0.2)^{0}}{(0)!} = 0.8187[/tex]
There is an 81.87% probability that the instrument does not fail in an 8-hour shift.
b. What is the probability of at least one failure in a 24-hour day?
The mean for an hour is 0.025 failures.
For 24 hours, we have [tex]\mu = 24*0.025 = 0.6[/tex]
Either we have at least one failure, or we have no failures. The sum of the probabilities of these events is decimal 1. So
[tex]P(X = 0) + P(X \geq 1) = 1[/tex]
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
In which
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-0.6}*(0.2)^{0}}{(0)!} = 0.5488[/tex]
So
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.5488 = 0.4512[/tex]
There is a 45.12% probability of at least one failure in a 24-hour day.
