Human genetics is a bit more complicated than that. There's not a single gene that determines hair color, but several. There's still some controversy over exactly which genes control what, but in a nutshell: hair color on the scale from pale blonde to dark black is controlled by one set of genes, and another gene controls the production of pigment that turns hair red. The latter gene is
dominant, but if the former genes result in dark hair, the red hair pigment is overwhelmed. So if a child is born to parents that would otherwise have fair hair but receives a gene from
either parent for red hair pigment, he will have red hair. Reference
here. This is why red hair tends to run in families, like mine, and also why it's not uncommon for nonredheaded parents to produce redheaded offspring, especially if one of them has fair hair, as Kimberley Thune does.
Red hair and blue eyes are more recessive genes than blond genes. I know there are variations. I was using the example of parents that have genes for purely dark eyes and hair and have purely recessive genes for red hair and blue eyes. To simplify the the discussion. You get AA(25%), AB + BA(50%), and BB (25%). John Thune and his wife had 4 to 1 odds against them with their 1st red-haired girl. And I believe 4^2 to 1 odds against them having a second red-haired girl. Am I wrong with the math? Is it 8 to 1 odds for the second child?