Generally, darker colours dominate so a person with one brown-eyed gene and one blue-eyed gene will have brown eyes. The only way to have blue eyes is to inherit two copies of the blue-eyed gene.
However, brown-eyed parents can pass a recessive blue-eyed gene, so two brown-eyed partners can both birth a blue-eyed baby.
If all that sounds a bit too confusing, then this handy chart should help answer all your questions. It's also worth remembering that a lot of babies are born with blue eyes, but they don't stay that way.
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