What diameter hose to fit on an airlock?

I went to Ace hardware and bought 1/4" and 1/2", I wasn't sure what size fits on the inner opening of the airlock, and apparently neither of them fit well enough. I'm a bit panicked because I was late for work and trying to get out the door with my daughter and what I had thought was a slow and steady fermentation has turned into something a little more. So I took the 1/4" hose, sanitized it and fit it into the stopper hole and that seemed almost a tight fit (don't like that) and I put the other end into an empty bottled water carboy, but I forgot to put sanitizer or water in it.
So 1) what OD or ID should I be looking for to fit onto the inner part of the airlock or are there better methods and 2) Why the need for water in the blow off container, and have I put my brew at great risk with no water/sanitizer at the end?

3/8" ID works for me.

Helps to dunk the hose end in hot water for a bit sometimes too.

1/2 " inner diameter, fits on the 3 pc airlocks I have.

I also use 1/2" on my 3-piece airlock.

Ok, I guess one of the hoses I bought will work, I thought it was half inch OD, but it says 7/16 and it fits inside the inner tube of the airlock perfectly. It's bubblying away in sanitizer and I wiped down the walls and everything within a foot of the carboy!