Enigma 1312: Hampshire v Middlesex
From New Scientist #2470, 23rd October 2004
Next season Hampshire and Middlesex will be among nine English counties in Division 1 of cricket’s league competition and also among the nine in Division 1 of the game’s county championship.
At the end of each season, at present, three counties are relegated to Division 2 of the league and three are relegated to Division 2 of the Championship, but it has been suggested that the number should be cut from three to two in each competition.
If we assume that in both competitions all counties have an equal chance of being relegated, what, as a fraction in its lowest terms, is the probability that in the season after next Hampshire and Middlesex will be in the same division as each other in both competitions if the number of counties relegated from Division 1 of each competition at the end of next season is (a) three, and (b) two?