Respuesta :
Answer:
a) The probability of not having failures in 8 days is 0.7866
b) The probability of having at least 1 failure in a 24-hour day is 0.5132
Step-by-step explanation:
a) Lets call X the number of failures in a 8 hour period. Since it is 8 hours insstead of 1, the mean will be 8*0.030 = 0.24. As a consequence, the probability that there are no failures in this period (in other words, X = 0), is
[tex]P_X(0) = \frac{e^{-0.24} * 0.24^0}{0!} = 0.7866[/tex]
b) Lets call Y the number of failures in 24 hours. Y has mean 24*0.03 = 0.72. The probability that Y is at least 1 can be computed by calculating the probability that Y is equal to 0 and substract from 1 this number (because obtaining 0 failures is the complementary event). The probability of Y being equal to 0 is
[tex]P_Y(0) = \frac{e^{-0.72} * 0.72^0}{0!} = 0.4868[/tex]
Thus, the probability of having at least 1 failure in a 24-hour day is 1-0.4868 = 0.5132
Answer:
(a) 0.7866
(b) 0.5132
Step-by-step explanation:
For a Poisson random variable, X, its probability is given by
[tex]P(X=r) = \dfrac{e^{-\lambda}\lambda^r}{r!}[/tex]
λ is the mean of the distribution and is given by
[tex]\lambda = np[/tex]
n is the number of hours in this question and p is the failure rate.
From the question, p = 0.030.
(a) The probability that the instrument does not fail in an 8-hour shift is the probability of no failure.
Here, n = 8. Then λ = 0.030 × 8 = 0.24
[tex]P(X=0) = \dfrac{e^{-0.24}\times0.24^0}{0!} = e^{-0.24} = 0.7866[/tex]
(b) The probability of at least 1 failure in a 24-hour day is the complement of no failure in 24 hours.
Here, n = 24. Then λ = 0.030 × 24 = 0.72
[tex]P(X=0) = \dfrac{e^{-0.72}\times0.72^0}{0!} = e^{-0.72} = 0.4868[/tex]
[tex]P(X>0) = 1 - P(X=0) = 1 - 0.4868 = 0.5132[/tex]
