Totals of a four-sided and a six-sided die from a sample space diagram: probabilities, a conditional probability and a points game with zero expectation

A fair four-sided die, numbered \(1\) to \(4\), and a fair six-sided die, numbered \(1\) to \(6\), are rolled together. The sample space diagram shows every possible total of the two scores. Let \(T\) be the total score.

1. Find[6]
   1. \(\mathrm{P}(T=7)\);
   2. \(\mathrm{P}(T>7)\);
   3. \(\mathrm{P}(T=8\mid T>6)\).
2. In a game, a player rolls both dice. The player wins \(4\) points if the total is \(7\) and wins \(2\) points if the total is greater than \(7\). Otherwise the player loses \(k\) points. Find the value of \(k\) for which the expected number of points that the player wins is zero.[7]