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A team with a bye to provincial semi finals has almost 3 times (2.8 to be exact) as good a chance of the AI as a team drawn in the preliminary round. Fairness demands that all teams play in a provincial quarter final. There are at least 3 options depending on counties' attachment to their provinces.
1. Move teams permanently from Ulster and Leinster to Connacht and Munster (as suggested already).
2. Let teams take turns in the 'wrong' province. Thus, each Ulster county would play in Connacht once every nine years.
3. Play the preliminary rounds as now. Let the winners progress to the quarters as now. Let the losers move to the quarters of the ' wrong province.

Out of 280 back door teams, 6 have won Sam. That's 2.2% success.
Out of 40 front door teams, 4 have won Sam. That's 10% success.
Front door teams are almost 5 times more successful than back door.

9 provincial quarter final winners (or byes) have won Sam
1 quarter final (or preliminary) loser has won.

6 prov semi final winners have won Sam.
3 semi final losers have won.

4 prov final winners have won Sam.
2 final losers have won.

Winning provincial championship matches has been a big advantage in the race for Sam.