Probability of At Least One Player Having All Cards of a Suit

In summary, the probability of at least one player having all the cards of one suit when dealing 52 cards to four players is (13C13 x 39C13 x 39C13 x 39C13) / (52C13 x 39C13 x 26C13 x 13C13). This can happen in four different ways, one for each suit, and the total number of possible combinations is 52C13 x 39C13 x 26C13 x 13C13. To find the probability, we divide the number of combinations where one player has all the cards of one suit by the total number of possible combinations.
  • #1
saubbie
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Homework Statement



If all 52 cards are dealt to four players, what is the probability that at least one player has all the cards of one suit?

Homework Equations





The Attempt at a Solution



This could happen for one of the four suits, so for the first card, there are 52C4 possibilities (where the C represents a combination), then for the next 51C12, then 50C11, etc. I don't feel as though this is correct, though. Any suggestions?
 
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  • #2


Hello,

Thank you for your post.

To find the probability that at least one player has all the cards of one suit, we need to find the total number of possible combinations of cards that can be dealt to four players and the number of combinations that result in one player having all the cards of one suit.

First, we know that there are 52 cards in a deck and each player will receive 13 cards. So, the total number of possible combinations of cards that can be dealt to four players is 52C13 x 39C13 x 26C13 x 13C13 (where the x represents multiplication).

Next, we need to find the number of combinations where one player has all the cards of one suit. This can happen in four different ways - one for each suit.

For the first suit, there are 13C13 combinations where one player has all 13 cards of that suit. For the remaining three suits, there are 39C13 combinations where one player has all 13 cards of that suit.

Therefore, the total number of combinations where one player has all the cards of one suit is 13C13 x 39C13 x 39C13 x 39C13.

To find the probability, we divide the number of combinations where one player has all the cards of one suit by the total number of possible combinations. So, the probability is (13C13 x 39C13 x 39C13 x 39C13) / (52C13 x 39C13 x 26C13 x 13C13).

I hope this helps! Let me know if you have any further questions.


Scientist
 

1. What is the probability of at least one player having all cards of a suit in a standard deck of cards?

The probability of at least one player having all cards of a suit in a standard deck of cards is approximately 0.2353 or 23.53%. This means that in a game with four players, there is a 23.53% chance that one of them will have all 13 cards of a particular suit.

2. How is the probability of at least one player having all cards of a suit affected by the number of players in a game?

The probability of at least one player having all cards of a suit increases as the number of players in a game decreases. For example, in a game with two players, the probability is 0.5 or 50%, while in a game with six players, the probability decreases to 0.1568 or 15.68%.

3. What is the probability of at least one player having all cards of a suit in a game of Texas Hold'em poker?

The probability of at least one player having all cards of a suit in a game of Texas Hold'em poker is approximately 0.0002 or 0.02%. This is because in Texas Hold'em, each player is only dealt two cards, making it highly unlikely for one player to have all five cards of a suit on the table.

4. Does the probability of at least one player having all cards of a suit change if multiple decks of cards are used?

Yes, the probability of at least one player having all cards of a suit increases if multiple decks of cards are used. For example, in a game with two players using two decks of cards, the probability is 0.6667 or 66.67%, while in a game with six players using two decks, the probability is 0.7821 or 78.21%.

5. Is it possible for more than one player to have all cards of a suit in a game?

Yes, it is possible for more than one player to have all cards of a suit in a game, but the probability of this happening is extremely low. In a game with four players, the probability of two players having all cards of a suit is only 0.0061 or 0.61%.

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