I might be able to explain that annals of mathematics solution a little more clearly. It appears there is an acknowledgement that the probability is dependant on the location of the 1st three darts.
In essence the closer the 1st 3 are together, the greater the probability of the 4th being in the same hemisphere.
Take the extreme cases for example.
1) If the 1st 3 darts land on top of each other, then the probability of the 4th landing on the same hemishpere is 100% - you could define any other part of the entire sphere as part of the same hemisphere
2) If the 3 darts land on a "great circle", then technically the 3 darts don't lie in any hemisphere, thus the 4th one cannot land in the same hemisphere
The answer therefore varies between 0% and 100% depending on the location of the 1st three, but there is an exception in (1), if the dart lies directly opposite the 1st 3.
