2009 AJC P2 Q10

Customers at a supermarket pay for their purchases either by cash or credit card. The probability of a randomly chosen customer paying by credit card is \(0.3\).

1. Find the probability that, on a randomly chosen day, there are more than \(6\) customers among the first \(20\) customers who pay by credit card and the fifth customer is the second customer who does so.[3]
2. Fifty samples of \(15\) customers each were taken. If there is a \(95\%\) chance that the average number of customers who pay by credit card per sample lies in the interval \(\left( 4,a \right)\), find the value of \(a\), giving your answer to \(3\) significant figures.[3]