N1997 P2 Q7 [Modified]

In a large population 35% of the people have blood group A. Specimens of blood from the first five people attending a clinic are to be tested. It can be assumed that these five people are a random sample from the population. The random variable \(X\) denotes the number of people in the sample who are found to have blood group A.

1. Explain what is a random sample in this context.
2. Find \(\mathrm P(X<4)\) correct to 3 decimal places.
3. Three such samples of five people are taken.
   1. Given that each of these three samples has more than three people with blood group A, find the probability that exactly two of the samples each have four people with blood group A.
   2. Find the probability that one of these three samples has exactly one person with blood group A, another has exactly two people with blood group A, and the remaining sample has more than two people with blood group A.
4. Explain the significance of the phrase ‘in a large population’ in the first sentence of the question.