Ms. Carol was always kind of ditsy. Maybe her mom used to tease her that her father was the milkman and he was black.
Wikipedia claims:
"According to Channing's 2002 memoir, when she left home to attend Bennington College in Vermont, her mother Adelaide informed Channing that her father George, who Channing had believed was born in Rhode Island, had actually been born in Augusta, Georgia. Adelaide claimed her husband's father was German-American and his mother was African-American. Channing's grandmother had moved with George to Providence, Rhode Island for his opportunities. According to Channing's account, her mother reportedly did not want [Channing] to be surprised "if she had a black baby"."
I have an account on Ancestry.com and can see the census records for her family. The 1860, 1870 and 1880 census all show the race of Ms. Channing's paternal grandmother (Clara Page) to be white.