2012 RVHS P2 Q10

1. For events \(A\) and \(B\), it is given that \(\mathrm{P}(B)=0.4\) and \(\mathrm{P}(\,\,A|B'\,\,)=0.15\). Find \(\mathrm{P}(\,\,A\cup B\,\,)\).[3]
2. In the last stage of a variety game show, the contestant aims to open a treasure chest containing the grand prize. The chest can be opened only when its two locks are unlocked. Each lock requires a unique key to be unlocked.
   The contestant is given up to three rounds to select the two correct keys among six different keys on a tray. For each round, he can only select one key which will be tried on both locks. The selected key of a particular round will be removed from the tray after that round.
   Show that the probability that a contestant wins the grand prize in the second round is \(\frac{\mathrm{1}}{\mathrm{15}}\).[1]
   Find the probability that
   1. the grand prize is won and the contestant selects the correct key in the first round,[2]
   2. the contestant selects the correct key in the first round given that he wins the grand prize.[2]

The variety game show runs for five days a week, over a period of \(60\) weeks. Find the probability that the mean number of shows per week for which the grand prize is won in the second round is more than \(0.4\).[3]