So, here's how power-ups work... Buying a power-up (unlocking it) gives a universal 25% for a power-up to spawn during a checkpoint (and then the power-up will randomly be one of the power-ups you currently have unlocked).
So you must unlock all 4 in order to have the best case scenario for power-ups to appear. Mind you, with level 1 of each power-up, you have a 100% chance
to have a power-up spawn EVERY checkpoint. This means on mountain cup you will get 2 power-ups, etc.
We have scaling % like this so you can't "cheat" the system by just getting a Boost power-up for ex, ignoring coins and shields, and have a 100% chance of getting boost every time down.
Hope this helps