How many states of the USA have no professional teams in the four major sports?

The 4 major sports being the NFL, MLB, NBA, and NHL, the count of states having no teams in any of these leagues is 25:

Hawaii, Alaska, Nevada, Idaho, Montana, Wyoming, New Mexico, Nebraska, South Dakota, North Dakota, Iowa, Arkansas, Mississippi, Kansas**, Alabama, Kentucky, Virginia, West Virginia, South Carolina, Delaware, Connecticut, Rhode Island, Vermont, New Hampshire, Maine.

This list includes states which have no team permanently based in a city in that state. Exceptions include The New England Patriots, who are based in Foxborough, Massachusetts but represent all states in the New England geographical area. It is also said that the Carolina Panthers (Charlotte, NC - NFL) and Carolina Hurricanes (Raleigh, NC - NHL) represent both North and South Carolina.

** Note that the Kansas City Royals and Kansas City Chiefs both play on the Missouri side of the border.

http://sports.yahoo.com/nfl/teams/
http://sports.yahoo.com/mlb/teams/
http://sports.yahoo.com/nba/teams/
http://sports.yahoo.com/nhl/teams/

There are 25 states that don't have a team in MLB, NBA, NHL or NFL, and 25 states that have at least one. These states have none:

Alabama
Alaska
Arkansas
Connecticut
Delaware
Hawaii
Idaho
Iowa
Kansas
Kentucky
Maine
Mississippi
Montana
Nebraska
Nevada
North Dakota
New Hamapshire
New Mexico
Rhode Island
South Carolina
South Dakota
Vermont
Virginia
West Virginia
Wyoming

The other 25 states have at least one:

  Arizona (4) Arizona Cardinals NFL, Arizona Diamondbacks MLB, Phoenix Coyotes NHL, Phoenix Suns NBA.
  California (16) LA Rams NFL, Anaheim Ducks NHL, Golden State Warriors NBA, LA Angels of Anaheim MLB, LA Clippers NBA, LA Dodgers MLB, LA Kings NHL, LA Lakers NBA, Oakland Athletics MLB, Oakland Raiders NFL, Sacramento Kings NBA, San Diego Chargers NFL, San Diego Padres MLB, San Francisco 49ers NFL, San Francisco Giants MLB, San Jose Sharks NHL.
  Colorado (4) Colorado Avalanche NHL, Colorado Rockies MLB, Denver Broncos NFL, Denver Nuggets NBA.
  Florida (9) Florida Panthers NHL, Jacksonville Jaguars NFL, Miami Dolphins NFL, Miami Heat NBA, Miami Marlins MLB, Orlando Magic NBA, Tampa Bay Buccaneers NFL, Tampa Bay Lightning NHL, Tampa Bay Rays MLB.
  Georgia (3) Atlanta Braves MLB, Atlanta Falcons NFL, Atlanta Hawks NBA.
  Illinois (5) Chicago Bears NFL, Chicago Blackhawks NHL, Chicago Bulls NBA, Chicago Cubs MLB, Chicago White Sox MLB.
  Indiana (2) Indiana Pacers NBA, Indianapolis Colts NFL.
  Louisiana (2) New Orleans Pelicans NBA, New Orleans Saints NFL.
  Maryland (3) Baltimore Orioles MLB, Baltimore Ravens NFL, Washington Redskins NFL.
  Massachusetts (4) Boston Bruins NHL, Boston Celtics NBA, Boston Red Sox MLB, New England Patriots NFL.
  Michigan (4) Detroit Lions NFL, Detroit Pistons NBA, Detroit Red Wings NHL, Detroit Tigers MLB.
  Minnesota (4) Minnesota Timberwolves NBA, Minnesota Twins MLB, Minnesota Vikings NFL, Minnesota Wild NHL.
  Missouri (3), Kansas City Royals MLB, St Louis Blues NHL, St Louis Cardinals MLB, Kansas City Chiefs NFL
  New Jersey (3) NJ Devils NHL, NY Giants NFL, NY Jets NFL.
  New York (8) Brooklyn Nets NBA, Buffalo Bills NFL, Buffalo Sabres NHL, NY Islanders NHL, NY Knicks NBA, NY Mets MLB, NY Rangers NHL, NY Yankees MLB.
  North Carolina (3) Carolina Hurricanes NHL, Carolina Panthers NFL, Charlotte Bobcats NBA.
  Ohio (6) Cincinnati Bengals NFL, Cincinnati Reds MLB, Cleveland Browns NFL, Cleveland Cavaliers NBA, Cleveland Indians MLB, Columbus Blue Jackets NHL.
  Oklahoma (1) Oklahoma City Thunder NBA.
  Oregon (1) Portland Trail Blazers NBA.
  Pennsylvania (7) Philadelphia 76ers NBA, Philadelphia Eagles NFL, Philadelphia Flyers NHL, Philadelphia Phillies MLB, Pittsburgh Penguins NHL, Pittsburgh Pirates MLB, Pittsburgh Steelers NFL.
  Tennessee (3) Memphis Grizzlies NBA, Nashville Predators NHL, Tennessee Titans NFL.
  Texas (8) Dallas Cowboys NFL, Dallas Mavericks NBA, Dallas Stars NHL, Houston Astros MLB, Houston Rockets NBA, Houston Texans NFL, San Antonio Spurs NBA, Texas Rangers MLB.
  Utah (1) Utah Jazz NBA.
  Washington (2) Seattle Mariners MLB, Seattle Seahawks NFL.
Wisconsin (3) Green Bay Packers NFL, Milwaukee Brewers MLB, Milwaukee Bucks NBA.

District of Columbia (3) Washington Capitals NHL - Washington Nationals MLB - Washington Wizards NBA.

http://commons.wikimedia.org/wiki/File:Map_of_USA_showing_state_names.png
https://en.wikipedia.org/wiki/Major_professional_sports_teams_of_the_United_States_and_Canada

Slighty outdated since Nevada has an NHL team. The franchise started in the 2017-18 season.

https://en.wikipedia.org/wiki/Vegas_Golden_Knights

Nevada also has an NFL team, as the Raiders began playing in Las Vegas in 2020.

https://en.wikipedia.org/wiki/Las_Vegas_Raiders