Answer:
(a) 0.7866
(b) 0.5132
Step-by-step explanation:
For a Poisson random variable, X, its probability is given by
![P(X=r) = \dfrac{e^{-\lambda}\lambda^r}{r!}](https://tex.z-dn.net/?f=P%28X%3Dr%29%20%3D%20%5Cdfrac%7Be%5E%7B-%5Clambda%7D%5Clambda%5Er%7D%7Br%21%7D)
<em>λ</em> is the mean of the distribution and is given by
![\lambda = np](https://tex.z-dn.net/?f=%5Clambda%20%3D%20np)
<em>n</em> is the number of hours in this question and <em>p</em> is the failure rate.
From the question, <em>p</em> = 0.030.
(a) The probability that the instrument does not fail in an 8-hour shift is the probability of no failure.
Here, <em>n</em> = 8. Then <em>λ</em> = 0.030 × 8 = 0.24
![P(X=0) = \dfrac{e^{-0.24}\times0.24^0}{0!} = e^{-0.24} = 0.7866](https://tex.z-dn.net/?f=P%28X%3D0%29%20%3D%20%5Cdfrac%7Be%5E%7B-0.24%7D%5Ctimes0.24%5E0%7D%7B0%21%7D%20%3D%20e%5E%7B-0.24%7D%20%3D%200.7866)
(b) The probability of at least 1 failure in a 24-hour day is the complement of no failure in 24 hours.
Here, <em>n</em> = 24. Then <em>λ</em> = 0.030 × 24 = 0.72
![P(X=0) = \dfrac{e^{-0.72}\times0.72^0}{0!} = e^{-0.72} = 0.4868](https://tex.z-dn.net/?f=P%28X%3D0%29%20%3D%20%5Cdfrac%7Be%5E%7B-0.72%7D%5Ctimes0.72%5E0%7D%7B0%21%7D%20%3D%20e%5E%7B-0.72%7D%20%3D%200.4868)
![P(X>0) = 1 - P(X=0) = 1 - 0.4868 = 0.5132](https://tex.z-dn.net/?f=P%28X%3E0%29%20%3D%201%20-%20P%28X%3D0%29%20%3D%201%20-%200.4868%20%3D%200.5132)