(a) The total number of students who play the guitar is \(44 + 38 = 82\).

The probability that a student plays the guitar is \(\frac{82}{180} = \frac{41}{90}\).

(c) Let \(F\) be the event that the student is female, and \(G\) be the event that the student plays the guitar.

\(P(F) = \frac{100}{180} = \frac{5}{9} \approx 0.5556\)

\(P(G) = \frac{82}{180} = \frac{41}{90} \approx 0.4556\)

\(P(F \cap G) = \frac{38}{180} = \frac{19}{90} \approx 0.2111\)

To check for independence, we compare \(P(F \cap G)\) with \(P(F) \times P(G)\).

\(P(F) \times P(G) = \frac{5}{9} \times \frac{41}{90} = \frac{205}{810} = \frac{41}{162} \approx 0.2531\)

Since \(P(F \cap G) \neq P(F) \times P(G)\), the events are not independent.