You are an engineer working for an air filtration system company. The company sells the air filtration system with a $1$ year warranty, which means the company will replace the unit at no cost if the unit is no longer effective within $1$ year of use.

Through extensive testing, the average life $($when the filter is no longer effective and needs to be replaced$)$ with an exponential distribution is determined to be $2$ years.

What is the probability that the product will be replaced under warranty?

Continuing from the previous question.

The business analyst has determined that if the company replaces $25$ out of $100$ units under warranty, the product will not be profitable. Suppose the probability that the product is replaced under warranty is $0\mathrm{.}35($note this is not the answer to the previous question$).$

What is the probability that the product will be not profitable? Use Table A$\mathrm{.}3$ as reference.

Hint: use normal approximation of the binomial distribution.