Discrete Random Variables (DRV)

In a large population, \(27.5\)% of the population has dimples on their cheeks. Samples of \(12\) people are randomly chosen from the population.

1. Calculate the probability that more than \(4\) people in a sample has dimples on their cheeks.
2. \(4\) such samples of \(12\) people are taken. Find the probability that
   1. Each of the \(4\) samples has more than \(4\) people who have dimples on their cheeks.
   2. One of the \(4\) samples has exactly \(3\) persons who have dimples on their cheeks, while the remaining \(3\) samples have more than \(3\) people who have dimples on their cheeks.
   3. Two of the \(4\) samples have less than \(4\) persons who have dimples on their cheeks, another has exactly \(4\) persons who have dimples on their cheeks and the remaining sample has more than \(4\) persons who have dimples on their cheeks.
   4. There are total of \(8\) people who have dimples on their cheeks from the \(4\) samples.