2010 AJC P2 Q9

In order to be offered a scholarship, a candidate has to pass two rounds of interview. (It is assumed that all interviewers’ decisions are independent.)

In the first round, there will be a panel of \(10\) interviewers and the probability of each interviewer passing a candidate is \(0.9\). The candidate fails to qualify for the second round if more than one interviewer decides not to pass him or her.

Find the probability that a candidate passes the first round of interview.

In the second round, there will be a panel of \(5\) interviewers and the probability of each interviewer passing a candidate is \(0.8\). The candidate is offered a scholarship only if all interviewers pass him or her in the second round.

1. Show that the probability that the candidate is offered the scholarship is \(0.241\), correct to three significant figures.
2. There are \(n\) candidates going for the interviews. Find the smallest \(n\) such that there is at least a \(98\%\) chance of \(2\) or more candidates being offered the scholarships.