Question

How many ways can you draw a club or a heart from an ordinary deck of cards? A spade or an ace? An ace or a jack? A card numbered 3 through 9 ? A numbered card (Aces are not numbered cards) or a king?

Short answer

26 for clubs/hearts, 16 for spade/ace, 8 for ace/jack, 28 for 3-9, and 40 for numbered/king.

Step-by-step solution

Understanding the Deck

An ordinary deck of cards consists of 52 cards which include 4 suits: clubs, hearts, spades, and diamonds. Each suit has 13 cards including numbered cards from 2 to 10, face cards (Jack, Queen, King), and an Ace.

Calculating Clubs or Hearts

There are 13 clubs and 13 hearts in a deck. To find the total number of ways to draw a club or a heart, add these two numbers together: 13 (clubs) + 13 (hearts) = 26 cards.

Calculating Spade or Ace

There are 13 spades and 4 Aces in a deck. Note that one of the Aces is already counted as a spade, so it is double-counted. Thus, use this formula: 13 (spades) + 4 (aces) - 1 (double-counted) = 16 cards.

Calculating Ace or Jack

For the ace or jack: there are 4 Aces and 4 Jacks. No intersection occurs here, so add them directly: 4 (aces) + 4 (jacks) = 8 cards.

Calculating Numbered Card 3-9

Each suit has one card numbered 3 to 9, making 7 cards per suit. There are 4 suits, so total: 7 (number range per suit) × 4 (suits) = 28 cards.

Calculating Numbered Card or King

Numbered cards range from 2 to 10, making 9 cards per suit across 4 suits: 9 × 4 = 36 cards. Add the 4 Kings: 36 + 4 = 40 cards.

Key concepts

Counting in Probability

Counting is a crucial skill when dealing with probability. It involves determining the number of possible outcomes for an event. For example, in card games, knowing how many ways you can draw a specific type of card helps in calculating probabilities. This is often done using techniques such as addition for mutually exclusive events and subtraction for overlaps. If you have two events, A and B, occurring simultaneously, the principle of inclusion-exclusion helps. It is expressed as \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), where \( \cap \) indicates the intersection or overlap between two events. By mastering counting, you can better assess the likelihood of different outcomes, setting a strong foundation for advanced probability topics.

Card Combinatorics

Card combinatorics refers to counting various combinations possible with cards, harnessed in games and probability questions. It includes understanding how sets of cards can be picked or assembled, while often considering constraints, such as specific suits or face cards. For instance, when calculating how many ways to draw a club or heart, we simply add because these are separate, mutually exclusive events: \(13 + 13 = 26\). If there's an overlap, such as counting spades and aces, the formula changes to account for this duplication. An example is the overlap in spades and aces, where we must subtract 1 double-counted card, leading to \(13 + 4 - 1 = 16\). Mastery of these principles allows for efficient solving of such combinatorial problems.

Deck of Cards

A standard deck of cards is foundational in many probability exercises and games. It consists of 52 cards divided into four suits: clubs, hearts, spades, and diamonds. Each suit contains 13 cards, which include numbered cards from 2 through 10, along with a Jack, Queen, King, and Ace. Understanding the composition of a deck is vital when calculating probabilities or managing outcomes. For example, knowing there are 7 numbered cards from 3 to 9 per suit helps in quickly calculating combinations. This translates to 28 cards (as in \(7 \times 4\)) for all suits. By grasping the structure and makeup of a deck, one enhances their analytical capabilities in both mathematical and strategic aspects of card games.