B727 CREW MADE APCH TO THE WRONG RWY AT MIA.

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VISUAL APCH TO RWY 9L INSTEAD OF RWY 9R. MIA APCH ADVISED US THAT WE WOULD BE ASSIGNED RWY 9L. WE REQUESTED RWY 9R. MIA APCH LATER ADVISED THAT RWY 9R WOULD BE APPROVED. WE WERE IN THE CLOUDS DURING THIS TIME. WHEN CLRED TO 3000 FT AND HDG 180 DEGS; WE ADVISED THE ARPT IN SIGHT. MIA APCH CLRED US FOR A VISUAL TO RWY 9R (I BELIEVE). I HAD A RWY IN SIGHT AND LINED UP VISUALLY FOR THIS RWY. MIA APCH CLRED US TO CONTACT MIA TWR; AND WE DID. I BELIEVE THE TWR CLRED US TO LAND ON RWY 9L; BUT I AM NOT CERTAIN. THE TWR ADVISED US OF TFC TAKING OFF ON THE RWY; AND WE SAW TFC TAKING OFF ON RWY 9L. THE TWR ADVISED OF TFC LNDG ON RWY 12 AND WE SAW THAT TFC LAND ALSO. I THEN REALIZED THAT I WAS LINED UP FOR RWY 9L; NOT RWY 9R. WE ASKED THE TWR TO VERIFY LNDG CLRNC ON RWY 9L; AND THE TWR REPLIED 'CLRED TO LAND RWY 9L.' WE DID NOT CREATE ANY CONFLICT WITH ANY OTHER ACFT -- IN THE AIR OR ON THE GND -- BUT I THINK WE WERE CLRED FOR A VISUAL APCH TO RWY 9R; AND I LINED UP ON RWY 9L AND CONTINUED THE APCH UNTIL LNDG. SUPPLEMENTAL INFO FROM ACN 458358: WE WERE CLRED FOR A VISUAL APCH TO RWY 9R. ON SHORT FINAL TO RWY 9L THE CAPT REALIZED WE WERE ON SHORT FINAL TO RWY 9L INSTEAD OF RWY 9R. HE WISELY REQUESTED CLRNC TO LAND ON RWY 9L AND RECEIVED CLRNC TO LAND FROM MIA TWR TO LAND RWY 9L.