Respuesta :
Answer:
a) 0.1217 = 12.17% probability that a disk has exactly one missing pulse
b) 0.0089 = 0.89% probability that a disk has at least two missing pulses
c) 0.7559 = 75.59% probability that neither contains a missing pulse
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.
In this problem, we have that:
[tex]\mu = 0.14[/tex]
(a) What is the probability that a disk has exactly one missing pulse? (Round to four decimal places)
This is P(X = 1).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 1) = \frac{e^{-0.14}*0.14^{1}}{(1)!} = 0.1217[/tex]
0.1217 = 12.17% probability that a disk has exactly one missing pulse
(b) What is the probability that a disk has at least two missing pulses? (Round to four decimal places)
Either a disk had at most 1 one missing pulse, or it had at least two. The sum of the probabilities of these events is 1. So
[tex]P(X \leq 1) + P(X \geq 2) = 1[/tex]
We want [tex]P(X \geq 2)[/tex]. So
[tex]P(X \geq 2) = 1 - P(X \leq 1)[/tex]
In which
[tex]P(X \leq 1) = P(X = 0) + P(X = 1)[/tex]
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-0.14}*(0.14)^{0}}{0!} = 0.8694[/tex]
[tex]P(X = 1) = \frac{e^{-0.14}*0.14^{1}}{(1)!} = 0.1217[/tex]
[tex]P(X \leq 1) = P(X = 0) + P(X = 1) = 0.8694 + 0.1217 = 0.9911[/tex]
So
[tex]P(X \geq 2) = 1 - P(X \leq 1) = 1 - 0.9911 = 0.0089[/tex]
0.0089 = 0.89% probability that a disk has at least two missing pulses
(c) If two disks are independently selected, what is the probability that neither contains a missing pulse?(Round to four decimal places)
Each disk has a 0.8694 probability of having no missing pulses.
Since they are independently selected,
[tex]P = (0.8694)^{2} = 0.7559[/tex]
0.7559 = 75.59% probability that neither contains a missing pulse
