If both the mother and father have brown hair... how can this couple have a child with red hair

It involves things called "recessive" genes, which means that there must be two copies of this gene in the same person for it to be expressed (so you need 2 red hair genes, for instance). If there is one recessive gene and a "dominant" one, the dominant one overrides the recessive one and is expressed.

So if both parents had one red and one brown gene, then they would both have brown hair while still carrying a red hair gene that they could pass on. If both of them passed on their red one (you only pass one of your 2 copies of any given gene), then the child would have red hair.

In reality, traits such as hair colour are actually determined by several genes working together, and the overall result depends on the combinations and interactions between all the genes. This allows all manner of unexpected results, depending on what genes each parent is carrying.

Both parents carry the recessive red hair gene, but their brown haired genes are dominant. The child got just the recessive red hair genes from the parents. Example Father: Br Mother: Br = Children 25% BB, 50% Br, 25% rr. The child just hit the 1 in 4 chance of having red hair.

Also: 1 in 8 children born in American hospitals have a blood type that is incompatible with the fathers and mothers blood type possibilities. Meaning the father isn't the biological father.

There is this old wives tail where if the child has red hair and the parents don't, they conceived the child when she was menstruating lol.

Ok but in all seriousness, genes does not stop at the parents. It goes back to grandparents, and some great grandparents. My nephew has green eyes, his parents both have brown. We have a family history of hazel eyes so it all dates back.

if both parents have a recessive red trait it would not show up in themselves.
but 1 out of 4 of their children -one average- would have red hair.

They both had a red hair recessive gene "hiding" behind their brown hair.

recessive genes

Both parents had a recessive gene for red hair

Throw back.