Gold or Silver: A Tale of Two Conditions

Two sealed lockboxes sit in front of you. Independently, each was loaded with a gold coin with probability $1/2$ and a silver coin otherwise. Answer both parts: (a) A referee who can see inside both boxes announces, "at least one box holds gold." What is the chance both do? (b) Instead, you open one box yourself and find gold. What is the chance the remaining box also holds gold?