Dealing cards to 4 people

Question

I and three friends are playing a game of Open Face Chinese Poker involving a standard deck of 52 cards. Player X got a Queen Fantasy land which means the next time the game is played, he will get the first 14 cards from the deck.

The question is this: Is there any difference in probability of getting a Flush between these 2 situations:

1. The first 14 cards are given first to player X, then the rest of the cards are distributed sequentially between the other players
2. The cards are distributed sequentially to the other players first, then player X receives the last 14 cards

I felt that it is different and situation 1 is more advantageous to player X, but I cannot prove it mathematically.

Answer 1

A deck of cards is simply a permutation of the cards in a row. It doesn't matter which 14 cards you give him (the top 14, the bottom 14, the middle 14, the ones in the first 14 odd places and so on) he has the same probability to get the same hand.

The idea of the proof is symmetry and the following is a great simple example: you draw one card of the deck. Does the probability of it being king changes if you draw the first card or ignore the first and take the second?

Answer 2

In general, once a deck of cards has been shuffled, any specific card is just as likely to be in one position as any other position.

After shuffling, it doesn't matter how the order of the cards in the deck is changed, or in what order the cards are dealt out to the players; any card is still just as likely to be in one position as any other.

The reason that, just before the deal, a deck of cards is typically cut by a non-dealer, and that the cards are dealt one at a time to the players in order, is simply to make cheating more difficult. In terms of the distribution of cards each player gets, it makes no difference.

P.S. for those that don't know the game, the advantage to getting all the cards at the beginning is that one can decide where to play them while having full knowledge of all of one's cards. Normally players get only a few of their cards at a time, and must play them before they get to see any others.