# What are the odds of players drawing a specific card in a round?

FIRST: Thanks for the advice on finding out how to ask my question. I revised my question, and here it is. I hope what I'm looking for is clearer.

I'm making a game and I need help figuring out the odds of something happening.

Every round is made up of 6 turns. (If four players are playing, there will be 24 turns in a round.)

The board is filled with colored squares. Each turn, you have a 20% chance of landing on a BLUE square, which means you will draw a blue card. There are 72 blue cards; 7 of those are "Level Up" cards. Drawn cards are discarded. Non-Level-Up cards are irrelevant to this.

If four players are playing, what are the odds that in a round, at least one player will level up? What if six players are playing?

I'm aware that the odds increase/decrease based on cards drawn, so I'm just looking for the base odds.

If there are 24 turns:

P(Any player Levels up) = P(A Player Levels Up on turn 1) + P(A player levels up on turn 2) + ... P(A Player Levels Up on turn 24)


What is the probability that a player levels up on his or her turn?

Assuming they only level up with a blue level up card, then they have to land on a blue square AND draw a level up card.

P(A player levels up) = .2*P(Draw a level up card)


What's the probability of drawing a level up card? This will change each turn because you are discarding drawn cards. The first turn it will be 7/72. If a level up card is drawn, then the next turn it will be 6/71. Otherwise, 7/71.

Hope this helps.