In the render result, I'd like to be able to see the DNA inside the football.
How can I make the football transparent so that the DNA is clearly visible inside?
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If you must use Blender Render, you can set up a glass like material by enabling transparency and than play with the settings.
I would suggest
It is absolutely vital to have something to reflect, so you must set up a scene, or at least an non uniform background &/or lightening, or your object won't look like if they were made of glass. Don't focus only on the single material but to the whole scene.
To do this in Cycles, choose cycles render and open node editor.
Create your base material for the football (this should be what it should look like if there wasn't any transparency on it.
Then mix this with a transparent shader, and make any necessary shader tweaks to give it the appearance you're looking for.
The cylinder is positioned inside the sphere, the cube behind it
Please keep in mind that if you prefer to use the Blender Internal Engine, that's perfectly fine, this is just a way to do it using Cycles.