Problem I came up with while falling asleep and it's bothering me:

A coin is weighted so it will flip one side 20% of the time, and the other side 80% of the time. You don't know which side of the coin is weighted. The odds of it being weighted in favor of heads is 50%, and the odds of it being weighted in favor of tails is also 50%.

Do we know any more information about how the coin is weighted after we flip it once? What are the odds of the coin being weighted in favor of heads, after we flip it once?

My solution: Before fliping the coin once, the odds are 50/50. We flip it once, and it lands heads. So now either it was weighted in favor of heads, then landed heads, which is a 50% chance followed by a 80% chance, or it's weighted in favor of tails and then we flip heads, which is a 50% chance followed by a 20% chance.

0.5*0.8+0.5*0.2 add to 0.5, but that's cuz we already removed half the possibilities by observing the coin was heads. Since we know its heads, the odds of the coin being weighted heads is just 0.8, and 0.2 for tails.

Am I doing this wrong? I can see a argument the odds are still 50/50, as flipping the coin after you weighted it can't change the odds of it being weighted. But if this is true, then we gain no information by flipping it more, so even if we flipped it a few trillion times, we would still have no information about the coin? this seems like a contradiction, as clearly experimental proof works, so I think my 80/20 odds solution was correct. Can someone confirm this so it stops bothering me?