Must Objects Which Undergo Fission Cease to Exist?

In cases of fission a thing or person splits into two things or persons. It is often thought that in cases of fission the thing or person which fissions could not be numerically identical with one result of fission rather than the other, on the grounds that it would be objectionably arbitrary for the thing or person to end up as one result of fission rather than the other. I show that this sort of argument fails. Two fission cases which receive special attention are split-brain transplants (where each brain hemisphere is transplanted into a different body) and monozygotic twinning.