2013 HCI P2 Q10

A bag contains \(15\) tokens that are indistinguishable apart from their colours. \(2\) of the tokens are blue and the rest are either red or green. Participants are required to draw the tokens randomly, one at a time, from the bag without replacement.

1. Given that the probability that a participant draws \(2\) red tokens on the first \(2\) draws is \(\frac{1}{35}\), show that there are \(3\) red tokens in the bag.[3]
2. Find the probability that a participant draws a red or green token on the second draw.[2]
3. Events \(A\) and \(B\) are defined as follows:
   \(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\) \(A\): A participant draws his/her second red token on the third draw.
   \(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\) \(B\): A participant draws a blue token on the second draw.
   Find \(\mathrm{P}(A \cup B)\).[2]
4. Determine if \(A\) and \(B\) are independent events.[1]