Armor in XCom doesn't work the way you describe.
The way damage works is:
1- Take the average damage of the weapon
2- multiply by 0 to 200% (or 50% to 150% with the TftD formula)
3- Apply the damage modifier of the armor
4- Substract armor
5- If anything is left, remove HP.
6- If HP was removed, also remove 10% of damage, rounded up, from the armor value.
Say you have a plasma pistol shooting at an operative in personal armor in the chest.
1- Plasma pistol deals 52 damage
2- you roll for the damage modifier, say you get 110%. The damage becomes 57 (1.1*52, rounded)
3- The Personal Armor's damage modifier is 100%, so 100%*57 = 57;
4- Personal armor has 50 armor in the front: 57-50 = 7
5- The operative takes 7 damage to its HP.
6- Since damage was applied, armor is reduced by: 10%*7 = 0.7, rounded up is 1. So the front facing of the soldier's personal armor now only protects for 49.
Don't worry, things do work as expected.
If the first few shots do no damage it means that their modified damage (from step 3) was less than the armor on that facing, so at step 4 the armor absorbed everything, taking no damage (and of course no hp damage).
Then for the last shot, the damage roll (step 2) was high, so the penetrating damage (step 4) was more than the operative's HP, so he died (step 5).
In the early game, with soldiers with low health and personal armor being fired upon by plasma pistols, it is rather common to either escape without harm (poor damage roll at step 2, meaning no penetrating damage) or die. There is only a narrow range of values that do damage but not enough of it to kill the operative.
