N2023 P2 Q5

A child’s toy has \(36\) slots, numbered from \(1\) to \(36\). A child puts a ball into the toy, the ball falls into one of the \(36\) slots and the child’s score is the number of that slot. The ball is equally likely to fall into any one of the slots.

Sadiq is investigating four possible events, \(A\), \(B\), \(C\), and \(D\), which are defined as follows.

\(A\): The score is odd.

\(B\): The score is even.

\(C\): The score is a multiple of \(3\).

\(D\): The score is a multiple of \(6\).

1. 1. State which pairs of the events, if any, are mutually exclusive.[1]
   2. Show that \(A\) and \(C\) are independent events, and state another pair of independent events.[2]

Sadiq notices that a ball has become stuck in the slot labelled \(36\), and so balls put into the toy are now falling, with equal probability, into one of only \(35\) slots, and the score can only be from \(1\) to \(35\).

2. 1. State which pairs of the four events, if any, are now mutually exclusive.[1]
   2. Determine whether \(A\) and \(C\) are now independent events.[1]