Hmm... I could even make a bet on which composers may an average person have most likely heard about at least once.
Dead, likely known: Mozart, Beethoven, J. S. Bach, Tchaikovsky, Vivaldi, Chopin.
Dead, long-shot: Stravinsky, Schubert, Liszt, Brahms, Verdi, Wagner, Puccini, Rachmaninov, Mahler, Händel.
Total: 16
Living, likely known: John Williams, Philip Glass.
Living, long-shot: Danny Elfman, Hans Zimmer, Penderecki.
Total: 5
I'd love to be totally underestimating the average person's knowledge, though.