Step 1 :Let the radius of the sphere be r and the radius of the cylinder be R. The height of the cylinder will be 2r.
Step 2 :Volume of sphere = \(\frac{4}{3} \pi r^3\)
Step 3 :Volume of cylinder = \(\pi R^2 (2r)\)
Step 4 :P(fly in cylinder) = \(\frac{\text{Volume of cylinder}}{\text{Volume of sphere}}\)
Step 5 :\(\frac{1}{4}\) = \(\frac{\pi R^2 (2r)}{\frac{4}{3} \pi r^3}\)
Step 6 :Solve for r: \(r = 2.44948974278318 R\)
Step 7 :\(\boxed{r = 2.44948974278318 R}\)