Pulling the goalie is a highly volatile move. Yes, pulling the goalie increases the chance of scoring the game tying goal, but the odds of the opponent scoring obviously go up tremendously with the goalie in net. I'm sure there's been numerous analytics performed on when statistically the best time is to pull the goalie, and it's around the 2-minute mark.
If down 2 pull him early...2-4 mins left
If down 1 the idea is that your normal 5v5 should be able to tie the game....
Obviously pulling the goalie gives the opponent a wide open net....so the longer you can hold off the better...which is why usually with 1-2 mins left and full possession of the puck
If the losing team can't get full possesion of the puck the goalie pull will be delayed ...so sometimes they can't get him out til 1 min left
Other factors play in as well, such as line changes (Home has last line change), and when your top players last got on/off the ice. Plus, if you pull at 3-4 minutes to go, you'll realistically need to have your second line and maybe even 3rd line out there at some point, and they may not be the best puck possession players you want in a 6 on 5 position.
Each coach is different, as is each situation, but blindly pulling the Goalie at 3-4 minutes to go has no data that supports more successful outcomes vs. 1 minute to go.
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