(a) The possible combinations to get a score of 4 are (1,3), (3,1), and (2,2). There are 3 favorable outcomes. The total number of outcomes when two dice are thrown is 36. Therefore, the probability is \(\frac{3}{36} = \frac{1}{12}\).

(b) The mean of a geometric distribution is given by \(\frac{1}{p}\), where \(p\) is the probability of success. Here, \(p = \frac{1}{12}\), so the mean is \(\frac{1}{\frac{1}{12}} = 12\).

(c) The probability that the first success occurs on the 6th trial is given by \(\left( \frac{11}{12} \right)^5 \times \frac{1}{12}\). Calculating this gives \(0.0539\) or \(\frac{161051}{2985984}\).

(d) The probability that \(X < 8\) is \(1 - \left( \frac{11}{12} \right)^7\). Calculating this gives \(0.456\) or \(\frac{16344637}{35831808}\).