2008 RJC P2 Q10

The probability that a hockey team wins any match is \(\frac{1}{2}\) and the probability that it loses any match is \(\frac{1}{6}\). Three points are awarded for a win, one point for a draw and no point for a defeat. The team plays four matches and the outcome of each match is independent of the other matches.

1. Show that the probability that the team has exactly one draw and exactly one defeat is \(\frac{1}{6}\).[1]

Find the probability that the team

2. wins the first match and goes on to win exactly one other match,[3]
3. wins exactly one match, given that it obtains four points.[4]