Betty will always win.
See how.
As there are 15 numbers, Alice will always be chossing last. Now, consider what will happen during the last 3 choose operations.
Lets say Alice chooses any number at the 3rd place from last . Now, there are 2 nos left.
Now, there is a possibility that Betty chosses a number due to which she losses or choose a number due to which Alice is forced to choose a specific number at last . Obviously she takes the second option due to which Betty either wins or draws the game but never loses.