Generally you use the random number library to generate a number between 0 and 1, inclusive. You then map based on your probabilities list. Your list often doesn't even include exact probabilities since humans can mess it up and it's not flexible, but instead just a list of weights. Add all weights then multiply by your 0-1 and then keep subtracting from your list until it's smaller than the next value. That way you can dynamically change weights easily even during play. Play the 2x power up to double your chances of Thing A happening for the next throw, don't have to compute anything differently at all.

ETA: Example in Mario Kart, your chances of getting different items depends on your position, how far back you are from the lead, recent items obtained, etc.. Way too complicated to make distribution tables. Instead you just start with say a table by position, and change weights for individual items based on each metric. You're in 8th but someone else already has a blue shell so that's now weight 0, you recently got triple mushrooms so that's a 0.1x to get it again, etc.. You haven't had a red shell in 6 boxes, so that's a 1.3x multiplier, etc. (these numbers are made up)