Hmmm. London Underground carries about 1.5 billion people reach year. In the last year for data I can find, around 80 ended up on the tracks. Only a few stations have any pbarrier between track and platform.
How about making a study, counting all metro stations noting whether they have platform screen doors, automation, and the year built. You'll very likely see there's a strong connection between automation and platform screen doors, but less so for the other variables.
Newer lines, like the new section of the Jubilee line have straight track and doors.