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(requires basic knowledge of Hypershade workflow)

Translucency, like subsurface scattering, is an important characteristic of many typed of surfaces. When an object is very thin, light is able to pas through, becoming scattered, and contributing to the illumination of the surface.
The standard ways of accomplishing translucency, using the built in controls, are notoriously unreliable, or inefficient. They are also difficult to fine tune.
There are MentalRay nodes which allow you to create efficient and easily controllable translucency.

Begin by setting up the shading for the surface, without translucency (how it looks on the illuminted parts.)

Hypershade Workflow

The most important is the node, misss_lambert_gamma. This is a lightmap node that typically is used with SSS shaders. It has the option for flipping normals. Set the flip to 1. If you do not wish to have the shader take indirect lighting, such as final gather, into account, uncheck Indirect lighting. Leave the other settings default for now.

Next create a misss_fast_shader node. If a lightmap and Mentalray texture get created automatically, you can delete them. Plug the lambert_gamma node into the diffuse shader of the fast shader. Set the SSS weights to 0. The diffuse color should be set to what ever the translucency color should be, which is usually a more saturated version of the surface color. If you are using a file or procedural texture for the original diffuse color, you might want to plug it into a Remap HSV node to saturate it.

The diffuse weight in the Fast Shader will determine how much the translucency has an effect.

The next step will depend on the type of shader you used in the beginning for diffuse.
For a standard Maya material, plug the output of the Fast Shader into the incandescence.
ForĀ  Mia_Material_x, plug the output into the additional color, in the advanced section.
An alternative is to create an add node (found in Maya utilities), and plug in the fast shader, and the surface material (for Material X, use the result as the output).

Take in to account other properties of the surface, like bump, which would likely affect the translucency. You can remap the bump texture, so areas that have positive relief are slightly darker than the rest, then use this as the diffuse weight value in the Fast Shader