Основываясь на быстром тестировании, я должен сделать вывод, что ORDER BY RAND () применяется только после применяется оператор WHERE, а не ко всему набору данных.
Результаты из таблицы с 50000 строк:
SELECT * FROM `mytable` LIMIT 1 (1 total, Query took 0.0007 sec)
SELECT * FROM `mytable` WHERE First = 'Hilda' LIMIT 1 (1 total, Query took 0.0010 sec)
SELECT * FROM `mytable` WHERE First = 'Hilda' (142 total, Query took 0.0201 sec)
SELECT * FROM `mytable` WHERE First = 'Hilda' ORDER BY RAND() LIMIT 1 (1 total, Query took 0.0229 sec)
SELECT * FROM `mytable` WHERE First = 'Hilda' ORDER BY RAND() (142 total, Query took 0.0236 sec)
SELECT * FROM `mytable` ORDER BY RAND() LIMIT 1 (1 total, Query took 0.4224 sec)