Answer
$P(E)=\frac{1}{15}\approx0.0667$
Work Step by Step
Consider for simplicity that:
A = The Screamin’ Eagle (wooden roller coaster)
B = The Boss (wooden roller coaster)
C = River King Mine Train
D = Batman the Ride
E = Mr. Freeze
F = Ninja
G = Tony Hawk’s Big Spin
H = Evel Knievel (wooden roller coaster)
I = Xcalibur
J = Sky Screamer
Event = "Ethan will ride two wooden roller coasters" = {AB, AH, BH}. So, $N(E)=3$.
Sample space: S = {AB, AC, AD, AE, AF, AG, AH, AI, AJ, BC, BD, BE, BF, BG, BH, BI, BJ, CD, CE, CF, CG, CH, CI, CJ, DE, DF, DG, DH, DI, DJ, EF, EG, EH, EI, EJ, FG, FH, FI, FJ, GH, GI, GJ, HI, HJ, IJ}. So, $N(S)=45$.
Using the Classical Method:
$P(E)=\frac{N(E)}{N(S)}=\frac{3}{45}=\frac{1}{15}\approx0.0667$