Math Problem Statement

What is the probability of drawing each of the following cards from a standard playing deck? See the entry “playing cards” in the glossary to learn what playing cards are included in a deck.

Solution

A standard deck of playing cards contains 52 cards, which are divided into four suits: Hearts, Diamonds, Clubs, and Spades. Each suit has 13 cards: Ace, 2 through 10, Jack, Queen, and King. Hearts and Diamonds are red suits, while Clubs and Spades are black suits.

Let’s go through the probability of drawing specific types of cards:

  1. Probability of drawing a red card:

    • There are 26 red cards in total (13 Hearts + 13 Diamonds).
    • The probability is: P(Red card)=2652=12P(\text{Red card}) = \frac{26}{52} = \frac{1}{2}
  2. Probability of drawing a face card (Jack, Queen, King):

    • There are 12 face cards in total (3 face cards per suit × 4 suits).
    • The probability is: P(Face card)=1252=313P(\text{Face card}) = \frac{12}{52} = \frac{3}{13}
  3. Probability of drawing an Ace:

    • There are 4 Aces in total (one in each suit).
    • The probability is: P(Ace)=452=113P(\text{Ace}) = \frac{4}{52} = \frac{1}{13}
  4. Probability of drawing a specific card (e.g., the Queen of Hearts):

    • There is only one Queen of Hearts in the deck.
    • The probability is: P(Queen of Hearts)=152P(\text{Queen of Hearts}) = \frac{1}{52}
  5. Probability of drawing a Spade:

    • There are 13 Spades in the deck.
    • The probability is: P(Spade)=1352=14P(\text{Spade}) = \frac{13}{52} = \frac{1}{4}

Would you like further details or have any questions?

Additional questions:

  1. What is the probability of drawing two red cards consecutively without replacement?
  2. What is the probability of drawing a face card or a red card?
  3. How do probabilities change if cards are drawn with replacement?
  4. What is the probability of drawing a number card (2 through 10) from the deck?
  5. How does the probability of drawing a King compare to that of drawing a face card?

Tip: When calculating probabilities, always ensure you know whether the events are independent or dependent (with or without replacement).

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Math Problem Analysis

Mathematical Concepts

Probability
Combinatorics

Formulas

P(Event) = (Number of favorable outcomes) / (Total possible outcomes)

Theorems

Basic Probability Theorem

Suitable Grade Level

Grades 6-8