The star is the big orange circle in the middle, and the filled blue circles show two extreme planet-orbit inclinations, above and below which the planet does not transit. Note that the orbit planes for the two configurations are parallel to the blue solid lines, not the black lines. The two orbit configurations are separated by and angle of approximately 2 Rstar/a (purple trace), obtained using the "skinny angle" property that the sine of a small angle is the small side over the long side.
With those definitions in mind, the transit probability is related to the solid angle traced out by the two extreme transit configurations, which is
as well as the total solid angle at a semimajor axis a, or:
The probability is the ratio of these two solid angles:
For more on all things transit, including eccentric orbits and other properties of the transit geometry, see Prof. Josh Winn's (MIT) excellent book chapter here:
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