The easy way if doing this is just using a physics engine and calculate the result depending on where the ball lands. Prototyping a game like that would take me a few hours probably.

That being said if the game asks for money to do anything it's highly unlikely that that's what it's doing, instead it's probably choosing your result based on how much do they want you to win and then calculate a path for the ball that gets this result.

Something that makes it difficult to do this second alternative is if you have some semblance of control, e.g. if you choose the strength to use for the ball or something, that makes it slightly trickier to pull off because you're against the clock on finding a path and might end up with an unrealistic move (however if those round things make the ball kick around like they do on a pin all machine they can be used to set the course you want).

Long story short it's easier to do the animation deciding the result but if the game asks for money in exchange for anything in-game it's more profitable to do the results deciding the animation.