What would have been the probability of Obama been born white (after all his mother was a white women)- and why do we consider him black?
for someone whose parents are white and black in appearance the chance of them "looking white" or "looking black" in terms of phenotype is near zero. that's because they are guaranteed of getting about half "white" and "black" appearance genes. consider skin color, there are many genes where whites and blacks differ totally. e.g,. SLC24A5, SLC452, TYR, etc. obama, would have one copy of each.
on the other hand, if obama was the child of two parents who were themselves first generation biracial, in theory he could look totally white, or totally black, with an expectation in the middle. that's because both parents are diverse and sampling could result in a child inheriting only "white" or "black" appearance genes from their parents.
as for why obama is considered black, that's a function of sociology. in the USA you inherit the status of your lower status parent, especially if you can't "pass" as white. in africa i assume many would consider obama white, or not all black, because he's obviously mixed-race in appearance.