The pancake you picked up from the hat has a brown side, for you are looking at a brown pancake. Therefore, it's either the brown/golden or brown/brown one. Let denote the former BG and the latter B1B2. You may be looking at any one of the three browns sides: B, B1, B2. In the first case, the other side is golden. In the other two cases, the other side is brown. Therefore, the other side is brown in 2 out of 3 cases.

This is actually the same problem as Lewis Carroll's for which there is a formal solution at the This and that forum.