In an n-dimensional hyperspace, how likely is it that a randomly chosen plane passing through the origin will intersect "all-positive co-ordinate space"?
(By "all-positive co-ordinate space" I mean the space where all cartesian coordinates are positive. For instance, in a 2-D space, this would be the upper right quadrant. In 3-D space, this would be the region where x>0 and y>0 and z>0.)
Equivalent question: in n-D space, given a randomly chosen vector, how likely is it that a perpendicular vector exists with all positive cartesian coordinates?
The hyperplane will intersect "all positive space" iff its normal vector is neither in "all positive space" nor in "all negative space". This gives a probability of $1-2^{1-n}$.