The three armies you get by default are not actually by default - it is actually calculated by the total number of territories you have, divided by three. If that number is less than three, it is rounded up to three - hence the default you start with. Any continent bonuses are added onto that.
Let's take your example: He has sixteen countries, including the continent of Australia. Armies per turn:
Total number of countries (16)
divided by 3:
= 5 (rounded down from 5.33) armies per turn
+ 2 armies for Australia continent bonus
= 7 armies per turn
Does that help?