In an automated filling operation, the probability of an incorrect fill when the process is operated at

a low speed is $0\mathrm{.}001.$ When the process is operated at a high speed, the probability of an incorrect

fill is $0\mathrm{.}01.$ Assume that $30\%$ of the containers are filled when the process is operated at a high speed

and the remainder are filled when the process is operated at a low speed.

$($a$)$ What is the probability of an incorrectly filled container?

$($b$)$ If an incorrectly filled container is found, what is the probability that it was filled during the highspeed operation?