2024 NYJC H1 Q10

In a particular college, an exchange programme qualification test involves two stages: a written test and an interview. To qualify for the exchange programme, a student must first pass the written test and then the interview.

• Students who fail the written test at the first attempt will be required to take a second attempt. Students are given only one attempt at the interview.
• The probability that a student passes the written test, on either attempt, is \(0.7\).
• The probability that a student passes the interview is \(0.4\).

1. Draw a tree diagram to represent this information.[3]
2. Show that the probability of a student qualifying for the exchange programme is \(0.364\).[2]
3. Find the probability that a student fails the written test in the first attempt, given that the student qualifies for the exchange programme.[3]

\(20\) students take the exchange programme qualification test. The results of all the students are independent of each other.

4. Find the probability that at least \(14\) of the students qualify for the exchange programme.[3]