Alcott was born in Kansas City, Missouri, and is Jewish. She won the U.S. Girls' Junior in 1973, She turned pro in 1975 at age 18, directly upon graduating from Palisades High School in Pacific Palisades, Los Angeles. Her first victory came in just her third start as a professional at the Orange Blossom Classic on the LPGA Tour. She went on to be named LPGA Tour Rookie of the Year. Alcott won four tournaments in a year three times, in 1979, 1980, and 1984. Her best year came in 1980, when in addition to those four victories she also won the LPGA Vare Trophy for lowest scoring average, finished second five times and was in the Top 10 in 21 out of 28 tournaments played.