Get list of students having actions in Oracle
This is simply an aggregation problem.

select
  student_id,
  count(case when type in ('Expulsion','Probation') and quarter=3-1 then 1 end) as previous,
  count(case when type in ('Expulsion','Probation') and quarter=3 then 1 end) as "CURRENT"
from student_actions
group by student_id;


See http://sqlfiddle.com/#!4/f4263/10/0