(I have an acoustic guitar, the Brand is "New York Pro")I know that when you change strings, you must make sure that that ball of the string is up against the surface inside the guitar, not up against the bridge pin that is suppose to hold it in. To make sure that it is like this, you must tug on the string after you push the bridge pin in so that the ball end of the string goes against the inside of the guitar and not the very end of the bridge pin if it pushes on the very end, then it will pop up)

My problem is that I am doing this, and it is working with all of my strings, except for the low e string. If I tug on the string after inserting the bridge pin, the string tightens upward, but it does not rest on the surface of the inside of the guitar. It is hanging in thin air, slightly bent, ready to push up against the bridge pin and pop it out. Even if I tug any more, the string won't get pulled any farther up. I think that problem has to do with the gauge of the string, but I am not really sure. I am using MARTINS ACOUSTIC SP PHOSPHOR BRONZE guitar strings, and the gauge is Medium .013-.056

I have tried to put the string in the hole to a point where the ball of the string was touching the bottom of the hole on the inside of the guitar, and I pushed the bridge pin in. This did not work because the bridge pin did not seem to fit in properly. I think it had to do with the fact that the last centimeter of the string, next to the ball end, was thicker than the rest of the string.

I watched a video on youtube a guy saying to bend each string a centimeter from the ball end before inserting into the hole to keep the ball from pushing up on the bridgepin. Whatever the natural mechanisms were, it seemed to work fine with every string except for the low string.

As I was restringing my guitar just now, I had my friend hold down the ball end of the e string on the inside of the guitar with his hand as I turned the tuning peg to tighten the string. I tuned the string, and I did not feel any tension and nothing popped out. Still, the bridge pin eventually popped out, and the effort that my friend put into hold down the ball end from the inside just bought it some time before it would happen again. I could tell that it was going to happen because the string was in air, between the guitar surface and the bridge pin, as always.

I'm think that maybe if I bend the very end of the string where the ball is it will stay in place. Would that work? I'm guessing that I would have to use a pair of super skinny needle nosed pliers. Still, is there an exact way of putting the string in where I can avoid all of this?

So, in every painstaking little detail, can you tell me how to install the low E string ball end into a hole so that the bridge pin won't pop out?
Cut the bottom of the bridge pin to make a 45 degree angle slope - that will stop the ball-end locating under the end of the pin. Works 10:10
A common complaint, we addressed this just a week or so again. Bend the ball-end of the string at an angle before inserting it into the hole, and make sure the bend points up towards the headstock.
This prevents the ball end from catching the end of the bridge-pin.
When you insert the bridge-pin, pull up on the string at the same time to make sure the ball-end is seated against the bridge-plate where it belongs, instead of the bottom of the pin.
Cutting the bridge pin as described above might achieve much the same effect, but it's not necessary.
BTW... Do this with all the strings, not just the bass ones.