1)When you say you are using the canonical definition of hemisphere, do you mean North and South? Or are you talking about East and West as well?
2) Out of curiosity, why 14 continents?
If you are using North and South as pre-defined hemispheres, ignoring East and West, I do think it can be reduced down to a coin flip if you want to prove that plates are unlikely to almost all be in the same hemisphere. This is because allowing them to overlap allows more clustering, not less.
12/14 continents is >80% and 11/14 is <80%.
If you drop a continent randomly, it has a 50/50 chance of being in the north or south.
The probability of 12 or more being in the northern hemisphere is 0.647%. The probability of 12 or more being in the southern hemisphere is 0.647%. Together, the probability of 12 or more being in the same N/S hemisphere is 1.294%
You can do the same for E and W, getting another 1.294%, for a total of 2.588% chance. This is allowing the continents to overlap, so the real chance would be lower.