A Deck of Cards

Overview

A standard deck of cards is commonly used in probability and combinatorics to illustrate concepts of counting and probability. It consists of 52 cards divided into suits and ranks.

Structure of a Deck

Suits:

• 4 suits: Hearts (♥), Diamonds (♦), Clubs (♣), Spades (♠).
• Hearts and Diamonds are red, Clubs and Spades are black.

Ranks:

13 ranks per suit: $2, 3, 4, \dots, 10, \text{Jack (J)}, \text{Queen (Q)}, \text{King (K)}, \text{Ace (A)}$

Special Groups:

• Face Cards: Jack, Queen, King (3 per suit, 12 total).
• Number Cards: 2 through 10
• Aces: Often treated as high cards but can be low in certain games.

Applications in Probability

Equally Likely Outcomes:

• Each card has an equal chance of being selected when shuffled properly.

Sample Space:

• A single card draw has 52 possible outcomes.

Event Examples:

• Probability of drawing a Heart:

$$P(\text{Heart}) = \frac{13}{52} = \frac{1}{4}$$

• Probability of a Face Card:

$$P(\text{Face Card}) = \frac{12}{52} = \frac{3}{13}$$

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Worked Examples

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Summary

• A standard deck has:
  • 52 cards, divided into 4 suits and 13 ranks.
  • Red (Hearts/Diamonds) and Black (Clubs/Spades).
  • Face Cards (12 total) and Aces (4 total).
• Useful for probability problems because each card is equally likely.
• Probabilities can involve:
  • Individual suits, ranks, or combinations.
  • Events with overlaps require subtraction of the overlap to avoid double-counting.