I'm willing to bet you do.
In fact, I have a hypothesis about this. I wholeheartedly believe these are the most well-known rap lyrics in the history of hip hop. I've lived four states and two countries and spent significant time in many of the major cities in North America, and never have I come across someone who doesn't know at least half of this.
Now, before posting tonight, I had a wonderful debate with my homie Marcel (who also gave me some revisions for the No Wack Verses Club). He reckons that "Rapper's Delight" by Sugarhill Gang is more well known, especially the first chorus and verse of the song.
The conversation eventually shifted towards the significance of hip hop culture in the shaping of The Fresh Prince of Bel Air's narrative. But I think the jury is still out on the original premise as to which song is more well known.
I'll say this. "Rapper's Delight" came out at a time when hip hop wasn't really heard outside of a 50 miles radius of New York. And yes, it was the first hip hop track to get substantial airplay, but its entrance into the collective consciousness didn't really happen until about 10 years ago. In fact, it took an old white lady covering "Rapper's Delight" in the 1998 film The Wedding Singer before mainstream America realized how much they'd been affected by this song's success.
Conversely, The Fresh Prince theme was a much more immediate influence on how American pop culture viewed hip hop. Many people who had very little exposure to hip hop culture were able to tune in to this show and witness a relatively accurate (though extremely non-threatening) sample of the hip hop perspective. That peek into something new and unknown is one of the foundational aspects that allowed the show to be so successful. People who had never known any rap songs at least knew the words to The Fresh Prince theme. That was their way in.
One day I'll figure out a way to test this scientifically. How would I do that?
P.S. I think Ms. Jackson by OutKast and Baby Got Back by Sir Mix-a-Lot are up there, too. People in Louisiana flying confederate flags play those at tailgating parties. Crazy.