Discrete Random Variables (DRV)

In a manufacturing plant, a sample of \(45\) items is randomly selected from a production batch. The batch will be rejected if there are more than \(2\) defective products and will be accepted if it has less than \(2\). If the number of defective products is exactly \(2\), another sample of \(35\) products will be randomly selected form the production batch, the batch is rejected if the \(2^{\mathrm{nd}}\) sample has any defective products, otherwise the batch is accepted. Given that the proportion of defective products is \(3\%\) of the total production at the plant.

1. Find the probability that the batch is accepted as a result of the \(1^{\mathrm{st}}\) sample.
2. Given that a \(2^{\mathrm{nd}}\) sample is selected from inspection, find the probability that the batch will be rejected.
3. Find the probability that the batch will be rejected.
4. It is given that \(0.75\) hours are required for the inspection of each product thoroughly. Find the expected number of hours required for the inspection of a batch at the plant.