A man with red hair (recessive) marries a woman with black hair whose
mother had red hair. What are the chances that their first child will
have red hair? Black hair?

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Request for Question Clarification by cynthia-ga on 11 Jun 2005 23:34 PDT

If you were asking about eye color, the answer would be a lot easier,
in fact, there's even an eye color calculator:

Eye Color Calculator
http://museum.thetech.org/ugenetics/eyeCalc/eyecalculator.html

Hair color is much more complex.  Read this:

Can you tell me more about the genetics of hair color? 
http://www.thetech.org/genetics/ask.php?id=39

At the least, we would need more information about the grandparent's hair colors.

~~Cynthia

The genetics of eye and hair color is complex and far more difficult
than the over-simplifications used in the teaching of Mendelian
genetics. My advice is that you shouldn't trust any eye-color
calculator.

If you are a student dealing with one of these oversimplified problems
at school, then:

Assuming that the allele conferring red hair color is not sex-linked,
the man would be r/r, same as the mother of his wife (also r/r).

Since the wife is has black hair, she must have at least a dominant
allele (let's call it R). Because her mother was r/r, she must also
have a recessive allele (r). So, the wife is R/r.

The progeny of tha man (r/r) and his wife (R/r), will be as follows:

One half R/r (those who inherit the R allele from the mother and an r
allele from the father), with b

As I was saying,

The progeny of tha man (r/r) and his wife (R/r), will be as follows:

One half R/r (those who inherit the R allele from the mother and an r
allele from the father), with black hair; and

One half r/r (those who inherit the r allele from the mother and an r
allele from the father), with read hair

So, the answer to your questions is 1/2