So, you have THREE coins and you pick ONE at random. Probability you have the double headed coin: 1/3.

Flip 4 times, all heads. Incidentally, since this is a prerequisite to the question (if it is observed that ...) the probability of this occurring is 100%. You learn nothing about the coin in your hand. Could have two heads, could be fair.

Probability your coin is the double headed one: 1/3.

It's been a long time since college, but I didn't even have to get my old textbook to figure that one out.