Nowadays, virtually all players use the Stayman Convention to explore for a possible 4-4 major-suit fit after partner has opened the bidding with one (or two) no-trump. ...
When declarer holds nine cards in a suit and is missing Q-x-x-x, the normal play is to cash the A-K to try to drop the queen. It is important to realize, though, that percentagewise the edge in favor of playing this way — as opposed to a second-round finesse — is very slight. For this reason, any clues acquired from the bidding or play may justify taking a finesse instead. ...
