The "odds" of finding a random shiny in a Max Raid Den is 1/4096. However these odds can be affected by multiple variables if you're shiny hunting.
If you are doing the date skip method, but skipping one day at a time, then the odds are ~1/16,384. This is because the game is coded so that every four days, a new Pokemon has a chance of being shiny. That chance is 1/4096 but because there are four Pokemon in four days the odds are ~ 1/16,384
If you are doing the date skip method, and skipping four days at a time, the odds are ~1/4096. This is because like I said above, the game only calculated a new shiny value every four days. Therefore the odds are ~1/4096.
It also should be noted that if you go into a Max Raid Den, and then say lose the raid, then the Pokemon you fight is the exact same Pokemon with the same stats, moves, ability EVs and IVs.
In addition, if you go to 18:55 in the video, Austin John talks about how the first three Pokemon, in the four day cycle will always be the same. This is of course talking about repeating the same four day cycle, but that is why you skip four days when shiny hunting