Only Angels Have Wings
(1939)

Carter (Cary Grant) runs an air mail delivery in the small South American port Barranca. If he goes 6 months without a missed delivery, he will get a lucrative contract. Bonnie (Jean Arthur) is on a ship making a short stop. She falls for one of the pilots, and stays in town. At first, she despises Carter, thinking he sends the men out to fly in dangerous weather, while he is safe. However, she sees that in the worst conditions, Carter refuses to send anyone else and flies himself.  The other pilot dies in a crash, and Bonnie becomes interested in Carter.

As the 6 month deadline approaches, Carter is losing pilots to illness and injury. Desperate for anyone to fly, he hires an unknown man called MacPherson.  Things get tense, as Mrs McPherson (Rita Hayworth) is Carter's old flame. Worse, MacPherson is a fake name, and he is really a pilot they hate for parachuting out of a plane and leaving Kid's brother to die in a crash.  With no other pilots, Carter has to keep MacPherson.  Bonnie continues pursuing Carter, but he seems to have no interest.

On the final day of the 6 months, weather is terrible, and thick fog blocks the mountain pass.  They have just obtained a new plane which they think can fly over the mountains and avoid the pass.  MacPherson volunteers to fly, but Kid (who can no longer fly because he is going blind) is eager to ride along.  Kid flips a coin with Carter to see if he can go.  Carter grabs the coin first and sees it has heads on both sides (which is how Kid won free drinks all along), but lets Kid go.  The plane hits a flock of large birds, losing an engine. A bird breaks the windshield and hits Kid injuring him badly. Kid tells MacPherson to parachute out, as there is no way to land the plane. MacPherson refuses, replying "that is what your brother said, and I never lived it down". MacPherson manages to crash land the plane. Kid lives just long enough to tell the others the truth about MacPherson's bravery.

Carter will have to make the final delivery himself. As he leaves, Bonnie asks if he wants her to be there when he gets back. He flips a coin, which comes up heads for her to stay. Furius that he would flip a coin, she storms off.  He gives her the coin, and flies off.  When she sees the coin had two heads, she realizes he wanted her to stay.  Her smile lets you know that Carter will make the flight, and she will be there waiting.

Thanks Joseph C!