Problem 36 - Algebra for College Students

Question

Solve each problem.A card is randomly drawn from a deck of 52 playing cards. Find the probability that it is a red card or a face card. (Jacks, queens, and kings are the face cards.) \(\frac{8}{13}\)

Step-by-Step Solution

Answer

The probability is \(\frac{8}{13}\).

Step 1: Identify Total Outcomes

There are a total of 52 cards in a standard deck of playing cards. This will be our denominator in calculating probabilities.

Step 2: Count the Number of Red Cards

In a deck, half the cards are red, which includes the hearts and diamonds suits. Thus, there are 26 red cards.

Step 3: Count the Number of Face Cards

Each suit has 3 face cards: the jack, queen, and king. Hence, there are 4 suits × 3 face cards per suit = 12 face cards in total.

Step 4: Identify Overlapping Cards

The overlapping cards are the red face cards. There are two suits that are red (hearts and diamonds), and each of these has 3 face cards: Jack, Queen, King. This gives us 2 × 3 = 6 red face cards.

Step 5: Use the Inclusion-Exclusion Principle

To find the probability of drawing a red card or a face card, we use: \[P( ext{red card or face card}) = P( ext{red card}) + P( ext{face card}) - P( ext{red face card})\] This translates to:\[\frac{26}{52} + \frac{12}{52} - \frac{6}{52} = \frac{32}{52}\]

Step 6: Simplify the Probability

Simplify \(\frac{32}{52}\) to its lowest terms by dividing the numerator and the denominator by their greatest common divisor, which is 4:\[\frac{32}{52} = \frac{8}{13}\]

Step 7: Conclusion

The probability that a randomly drawn card is either a red card or a face card is \(\frac{8}{13}\).

Key Concepts

Inclusion-Exclusion Principle

In probability, when we want to find the probability of either event A or event B happening, we often need to account for any overlap between the two events. This ensures we do not count these overlapping outcomes twice. This is where the Inclusion-Exclusion Principle comes into play. It allows us to find probabilities correctly when events are not mutually exclusive.

The formula for the principle when applied to the probability of two events, A and B, is:

• \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)

In the context of a card game, if event A is drawing a red card and event B is drawing a face card, we use this principle to subtract the overlapping probability of drawing a red face card. The reason is simple—red face cards are counted once among the red cards and once among the face cards, so we must subtract these to find the correct probability. This step guarantees we accurately measure the probability of either event happening without duplication.

Deck of Cards

A standard deck of playing cards is a common tool used in probability exercises and games. Understanding its structure is essential for calculating probabilities accurately. A typical deck contains 52 cards divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards number 1 through 10, plus the face cards, which are the jack, queen, and king.

• Hearts and Diamonds: These are red suits, totaling 26 cards.
• Clubs and Spades: These are black suits, also totaling 26 cards.

Knowing this breakdown helps in quickly identifying the number of cards that fit particular criteria, such as calculating how many cards are red or determining the number of face cards, which plays a crucial role in solving probability problems involving decks of cards.

Face Cards

Face cards in a deck are the jack, queen, and king of each suit. They are often considered special because, unlike numbered cards, they do not have a numerical value tied to them in the context of card games or probability counting. In each of the four suits—clubs, diamonds, hearts, and spades—there are three face cards. This adds up to a total of 12 face cards in a 52-card deck.

• Each suit contributes 3 face cards.
• The total in the deck is 4 suits \( \times \) 3 face cards = 12.

Understanding the proportion of face cards is vital when calculating probabilities. For example, knowing there are 12 face cards out of a total of 52 cards or focusing solely on red face cards—which would be 6 in total from the two red suits—illustrates how integral these cards are to probability calculations and the application of principles such as inclusion-exclusion.