2020 HCI P2 Q8

A set of \(9\) cards are numbered \(1\), \(1\), \(2\),\(3\), \(3\), \(4\), \(5\), \(5\) and \(6\). A round of matching game is played where two cards are drawn, one at a time without replacement. A match occurs when the two cards drawn have the same number.

1. Given that the sum of the numbers on the two cards is less than \(5\), find the probability of a match occurring in a game. Hence, or otherwise, determine if having a match is independent of having a sum of the numbers on the cards less than \(5\) for a game.[3]
2. Dave decides to try his luck at this game repeatedly until he obtains a match. What is the least number of rounds he will have to attempt such that the probability of obtaining a match is at least \(75\%\)?[4]
3. Instead of drawing only two cards in a round, Dave continues to draw the cards until he obtains a match. He draws one card at a time without replacement. Find the probability that a match occurs on his third draw.[2]