Choose by bottleneck, not a fake ratio
Throw a Coin officially describes two upgrade directions: Luck and Coins. Luck is directly described as a way to get better loot, while upgrading coins is part of the same fountain progression loop. The selected official sources do not provide costs, level scaling, probabilities, or a fixed ratio between them.
That means a claim such as โalways buy three Luck levels for every Coin levelโ would be speculation. Instead, read the live upgrade panels, decide which part of your loop is limiting progress, buy one observable improvement, and compare the next results.