Probability - Conditional Probability

At a factory that employs many workers, \(30\)% of the workers are members of the trade union \(A\), \(60\)% are members of the trade union \(B\), and \(10\)% are not members of any trade union. A sample of five workers is selected from the workers of the factory.

1. Determine the mean and standard deviation for the number of workers in the sample who are members of \(A\).[2]
2. Find, correct to three significant figures, the probability that the sample contains not more than \(2\) members of \(A\).[2]
3. Given that exactly \(4\) workers in the sample are members of trade unions, find the probability that the majority of the workers in this sample are members of \(A\).[4]