2012 SAJC P2 Q13

On the average, \(30\%\) of the students in Calculus Madness Institute (CMI) could do the Differential Equation question in Block Test Two. The principal randomly select a class of \(30\) students to analyse the results.

1. State, in the context of this question, two assumptions needed to model the results by a binomial distribution.
2. Find the probability that at least \(6\) students in the class could do that question.
3. Find the probability that only \(2\) students among the first \(8\) selected students in the class could do the question given that at least \(6\) students could do that question.
4. The probability of no more than \(5\) students could do the question in a randomly selected class exceeds \(0.9\). Find the largest possible number of students in that class.

\(50\) classes with \(30\) students each are selected randomly. Find the probability that an average of at least \(10\) students per class could do the question.