Winning Probability in Card Game: 1/4 + (13/51)?

In summary, if you have a card game where the first step is to choose a suite, and if the card drawn matches the chosen suite, the player wins. If the card drawn does not match the suite, the game continues and the player chooses another card without replacing the first card. If this second card matches the suite, the player wins. If the second card has not matched the suite either, the game is over and considered a loss. The probability of winning is the probability of success on the first card plus the quotient of the probability of failure on the first card, times the probability of success on the second card.
  • #1
dsrunner
1
0
I have a situation that seems easy, I just keep confusing myself. Say there's a card game in which the first step is for the player to choose a suite. After the suite is chosen, they draw a card. If the card matches their suite, they win and the game is over. If the card does not match the suite, the game continues and the player chooses another card without replacing the first card. If this second card matches the suite, they win. If the second card has not matched the suite either, the game is over and considered a loss. What is the probability of winning? Would it simply be the probability of the first card's success probability plus the second card's success probability: (1/4)+(13/51)? Or would the winning probability be the probability of success on the first card plus the quotient of the probability of failure on the first card, times the probability of success on the second card: (1/4) + (3/4)(13/51)?
 
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  • #2
Very simple, since you are not replacing the cards that are drawn, the deck is fat with the cards that have not been drawn. The more cards drawn of a given suit, the less probable that suit will be the next drawn. Crunch the numbers and you will find by the time you get to the 52 card in the deck [assuming a poker deck], the odds are %100 it will be the suit from which only 12 cards have been drawn. The odds are strictly determined by how many of each suit remain in the deck. The suit of the last card drawn is irrelevant.
 
  • #3
Let's take a quick look at an extreme case. What happens if you have a deck with two cards, an ace of hearts, and an ace of spades.
Now, your chance of winning should be 1 if you pick hearts or spades, right?
But, if you take (1/2) + (1/1) = 3/2 you get a number bigger than one, so that's incorrect.
Using the 'quotient' version will give you the correct answer.
 

What is the formula for calculating winning probability in a card game?

The formula for calculating winning probability in a card game is 1/4 + (13/51). This formula takes into account the probability of getting a desired card (13 out of 52 in a standard deck) and the probability of getting a card from the remaining deck after the first card has been drawn (13 out of 51).

How do you determine the probability of getting a desired card in a card game?

The probability of getting a desired card in a card game can be determined by taking the number of desired cards and dividing it by the total number of cards in the deck. For example, in a standard deck of 52 cards, the probability of getting a specific card is 1/52.

What does the fraction 1/4 represent in the winning probability formula?

The fraction 1/4 in the winning probability formula represents the initial probability of getting a desired card on the first draw. This is because there are four suits in a deck of cards, so the probability of getting a specific suit is 1/4.

Why is the probability of getting a card from the remaining deck 13/51 in the winning probability formula?

The probability of getting a card from the remaining deck in the winning probability formula is 13/51 because after the first card has been drawn, there are 51 cards left in the deck. Out of these 51 cards, there are still 13 cards of the desired suit left, so the probability of getting a desired card is 13/51.

Does the winning probability formula apply to all card games?

The winning probability formula 1/4 + (13/51) can be applied to most card games, as long as the game involves drawing cards from a standard deck. However, some card games may have different rules or use a different type of deck, so the formula may not apply in those cases.

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