Math

$P[\text{winning}]=0.25\\ P[\text{playing at home}]=0.8\\ P[\text{winning AND playing at home}] = 0.2\\ P[\text{winning}]P[\text{playing at home}] = (0.25)(0.8) = 0.2\\P[\text{winning AND playing at home}]= P[\text{winning}]P[\text{playing at home}] \\ \text{so the two events are independent}$

now that you've been shown the method apply it to (b)