My friends and I gathered to play a card game the other night. More than 15 people showed up, therefore we had to shuffle two decks of cards so everyone could play at the same time. The game generally begins by handing each player 5 cards, revealing one card on the ground and put the remaining cards facing down on the ground. Each player who cannot play or simply don’t want to according to what is revealed, should draw a card from the ground.
Here is the twist: The table we were sitting around were a huge one, someone suggested to split the unrevealed portion of the cards in two after shuffling it so players can access it more easily where they need to no matter where they sit.
And here is the question: In my opinion, it is not fair to split the cards on the ground, cause what if I needed a 4 to win the game and all 4s were in the half that was not mine to pick from. This is just an initial thought without any proof to back it up.
Can someone explain if I’m right or wrong? Can this person help me prove it?
The challenge here is to decide what "fair" means. If it means "My chances of winning in the mixed-deck game should be the same as in the single-deck game," then your argument is certainly right, because sometimes all the 4s will be in the other pile. But equally often, they'll all be in your pile, and that's a good basis for thinking about a different notion of fairness.
Here's an alternative notion: the game is "fair" if no one would choose to switch positions with anyone else. That amounts to saying that it's "fair" of the odds of winning don't depend on where you're sitting. But when does the position-switching question get asked? If it's "after the cards have been dealt," then even the original game isn't "fair". because if you get dealt a rotten hand, you'd happily ask to switch with anyone else in the game, thinking "NO one can possibly have a worse mess than this!"
So the "swap places" question has to be asked before the deal. And I think that if you look at the current setup --- each player gets to draw from one of two piles --- you can see that everyone is equally disadvantaged in a randomly-dealt game (or over all possible randomly-dealt games).
If you're playing on a long oval table, and the folks at the ends have to use "their" pile, but folks in the middle can draw from either pile, then there is an unfairness...but I don't think that's what you were talking about.
It's useful to think about how we select a person for a prize or a task --- we "draw straws" --- one person holds several straws in their hand, all the same length except one -- and the others each draw a straw, leaving the 'dealer' with the last one. The person who draws the short straw wins (or loses). Now the loser may say "no fair!", but they actually had the same chance of getting the short straw as anyone else --- they only call it unfair when it turns out to be them. In short: fairness in games is probably best defined as "equal chance of winning at the start" rather than "everyone ends up with the same score". :)