Answer to Question #315696 in Statistics and Probability for Mirylle Anne

Question

The number of failures of a testing instrument from contamination particles on the

product is a Poisson random variable with a mean of 0.02 failure per hour.

a) What is the 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 shift?

Answer 1

We have a Poisson distribution,

$\lambda=0.02; P_t(X=k)=\cfrac{(\lambda t)^k\cdot e^{-\lambda t}}{k!}=\cfrac{(0.02t)^k\cdot e^{-0.02t}}{k!}.$

(a) $P_8(X=0)=\cfrac{(0.02\cdot 8)^0\cdot e^{(-0.02\cdot8)}}{0!}=0.8521.$

(b)

$P_{24}(X\geq1)=1-P_{24}(X<1)=1-P_{24}(X=0)=\\ =1-\cfrac{(0.02\cdot24)^0\cdot e^{(-0.02\cdot24)}}{0!}=\\=1-0.6188=0.3812.$