So based on memory I thought typically only one team that got promoted to the premier league immediately dropped back down to the championship. So I decided to compile a list of all promoted and relegated teams starting in 2001-2002. I chose only this date range because this is as far back as soccernet.com goes and I didn't want to have to do any more research.
Based on these seven seasons three of those only had a team returning back to the championship after only one season in the premier league. So based on my assumption that the majority of promoted teams stay more than one year in the premier league, I was wrong, because four of the seven seasons had two recently promoted clubs dropping back down. You will also notice in the last seven seasons, that there has not been a case of all three promoted teams finishing in the bottom third the following year. Although Sunderland was close to pulling this off in the 2007-2008 season.
I used red to signify teams being relegated after one season
2001-2002 Promoted Teams
Manchester City
West Brom
Birmingham
2002-2003 Relegated Teams
West Ham
West Brom
Sunderland
2002-2003 Promoted Teams
Portsmouth
Leicester City
Wolverhampton Wanderers
2003-2004 Relegated Teams
Leicester City
Leeds United
Wolverhampton Wanderers
2003-2004 Promoted Teams
Norwich City
West Brom
Crystal Palace
2004-2005 Relegated Teams
Crystal Palace
Norwich City
Southampton
2004-2005 Promoted Teams
Sunderland
Wigan Athletic
West Ham United
2005-2006 Relegated Teams
Birmingham
West Brom
Sunderland
2005-2006 Promoted Teams
Reading
Sheffield United
Watford
2006-2007 Relegated Teams
Sheffield United
Charlton Athletic
Watford
2006-2007 Promoted Teams
Sunderland
Birmingham
Derby County
2007-2008 Relegated Teams
Reading
Birmingham
Derby County
2007-2008 Promoted Teams
West Brom
Stoke City
Hull
2008-2009 Relegated Teams
Newcastle United
Middlesborough
West Brom
2008-2009 Promoted Teams
Wolverhampton Wanderers
Birmingham
Burnley
2009-2010 Relegated Teams
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Prospect