2006 HCI P2 Q28 [Modified]

Andrew plays a game with 4 cards numbered 1, 2 , 3 and 4. In each round, a random sample of \(n\) cards is taken without replacement, where \(1\le n\le 4\) and there is equal chance of Andrew taking 1, 2, 3 or 4 cards. Let \(L\) denote the smallest value of the chosen cards. The distribution table for \(L\) is as follows:

1. Show that \(b=\frac{5}{24}\).
2. Show that \(\mathrm{Var}\left( L \right)=\frac{1895}{2304}\).
3. Andrew invites Jasmine to join in the game. In each round, only 1 player plays and the cards are replaced before the other player plays the game. Jasmine has the first move in the game. The first player to obtain the results \(L=1\) will win the game. Find the probability that Jasmine wins the game.