2025 ASRJC P2 Q11

1. In a survey conducted by a library, a group of \(100\) members were asked the number of days in a month they visit the library. The results are shown below:

A member is selected at random from the group. The events are defined as follows:
\(A\): The member visits the library on \(4\) days or less in a month.
\(B\): The member visits the library on \(2\) days or more in a month.
\(F\): The member is a female.
Find the following probabilities in terms of \(n\).

1. \(\mathrm{P}(A \cup B')\)[1]
2. \(\mathrm{P}(F | A')\)[2]
3. Given that \(\mathrm{P}(A \cap B) = \frac{3}{10}\), find the value of \(n\). Hence determine if \(A\) and \(B\) are independent, justifying your answer.[3]

2. A librarian is organising a large community reading event. To manage registration, he needs to create \(4\)-letter access codes for the participants using the letters from the word BOOKKEEPER.
   1. How many access codes can be formed if all \(4\) letters are distinct from one another?[1]
   2. Find the total number of access codes that can be created given that there is no restriction.[3]

To run an event, the librarian plans to select a team of \(12\) volunteers from a group of \(18\). The volunteers comprise three age groups: \(5\) youths, \(6\) young adults, and \(7\) seniors.

3. In how many ways can the team be formed if at least one volunteer must be chosen from each age group?[3]