(a) The mean of a geometric distribution with success probability \(p\) is given by \(\frac{1}{p}\). For a fair die, the probability of rolling a 5 is \(\frac{1}{6}\). Thus, the mean is \(\frac{1}{\frac{1}{6}} = 6\).

(b) The probability that a 5 is first obtained after the 3rd throw but before the 8th throw is calculated as:

\(\left( \frac{5}{6} \right)^3 \cdot \frac{1}{6} + \left( \frac{5}{6} \right)^4 \cdot \frac{1}{6} + \left( \frac{5}{6} \right)^5 \cdot \frac{1}{6} + \left( \frac{5}{6} \right)^6 \cdot \frac{1}{6}\)

Alternatively, it can be calculated as:

\(\left( \frac{5}{6} \right)^3 - \left( \frac{5}{6} \right)^7\)

This results in approximately 0.300.

(c) The probability that a 5 is first obtained in fewer than 10 throws is:

\(1 - \left( \frac{5}{6} \right)^9\)

This results in approximately 0.806.