They agree to first open A. Then, for Alan, if not the car he will open B. And if the game continues, Becky will first open A and then,if not the keys, will open C.
So if a goat is behind A (p=1/3), then Alan will open B and if a car the game will continue. Becky will open A and find the goat, then will open C and find the keys.
If there is a car behind A (p=1/3), Alan will find the car. The game continues. Becky opens A, then opens C. The keys will be behind C with p=1/2.
If the keys are behind A (p=1/3), Alan will then open B and find the car with p=1/2. The game continues. Becky opens A and finds the keys.
So the overall probability is 1/3 + 1/3 x 1/2 + 1/3 x 1/2 = 1/3 + 1/6 + 1/6 =2/3.