Can two brown-eyed parents can have a blue-eyed child?

Yes, they can, and this actually follows the most simple rules of genetics. Blue eyes are, generally speaking, recessive to brown eyes. Say “B” is a gene for brown pigment and “b” is a gene for no pigment (default blue). Now, everyone has two genes for each trait, one from Mom and one from Dad. So people who have the gene combination BB or Bb have brown eyes, because their cells are making at least some brown pigment. Only people who have genes bb have blue eyes. (This is a simplification, but it works for our purposes.) Suppose that the two parents have:

Man = Bb

Woman = Bb

Their child receives one gene from each parent, and therefore could be BB, Bb, or bb. So you see that under these circumstances, they could easily have a blue-eyed baby!

Answer provided by Dr. Universe
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