I'll have to look for a decent on-line source, but ...
Obama isn't much African-American. He's 1/8th or at most 1/4 African. Obviously 1/2 Caucasian. Then 1/4 or 3/8ths Arab.
What Randall said, Peter. He self-identifies as African American and he has the physical characteristics. So while someone can nitpick a pedigree and more power to them, if the person they're talking about says "I'm African American." chances are real good I'm going to head that direction rather than the other.