Discrete Random Variables (DRV)

In a vegetable farm, a sample of \(35\) produce is randomly selected from a production batch. The batch will be rejected if there are more than \(3\) rotten produce and will be accepted if it has less than \(3\). If the number of rotten produce is exactly \(3\), another sample of \(25\) produce will be randomly selected form the production batch, the batch is rejected if the \(2^{\mathrm{nd}}\) sample has more than \(1\) rotten produce, otherwise the batch is accepted. Given that the proportion of defective produce is \(8\%\) of the total produce at the farm.

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 \(0.15\) 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 farm.