According to wikipedia; The highest concentration of Cohen Modal Haplotype outside of Jewish Kohanim is amongst Yemenites. There is no reported data on Saudi Arabia ( I e-mailed a Saudi University about this a few months ago but did not get a reply.) Assuming Jewish Kohanim have authentic tradition, all descendants of Abraham will have a Cohen Modal Haplotype; as do the Samaritan Israelites. However, 95% of (non-Cohen) Jews do not have the CMH and are thus children of Converts.
I think it is highly unlikely that a blood descendant of Abraham would totally accept Judaism without receiving it through patrilineal sources.