Problem 21 - Precalculus with Limits

Question

In Exercises 21 - 24, find the probability for the experiment of selecting one card from a standard deck of \( 52 \) playing cards. The card is a face card.

Step-by-Step Solution

Answer

The probability of selecting a face card from a standard deck of 52 playing cards is \( \frac{3}{13} \).

Step 1: Identify the total number of outcomes

In a standard deck of 52 playing cards, we have a total of 52 possible outcomes.

Step 2: Identify the favorable outcomes

There are 12 face cards in a standard deck of 52 playing cards. These face cards (King, Queen, and Jack from each of the four suits - hearts, diamonds, clubs, spades) are our favorable outcomes.

Step 3: Calculate the probability

The probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. In this case, the probability \( P \) of selecting a face card is \( P = \frac{12}{52} \). Simplifying this fraction to its lowest terms results in \( P = \frac{3}{13} \).

Key Concepts

Standard Deck of Cards

A standard deck of cards consists of 52 distinct cards commonly used for various card games. Each deck is divided into four suits: hearts, diamonds, clubs, and spades.
Each suit contains 13 cards, making up the 52 cards in total. These 13 cards in each suit are as follows:

• Numbers from 2 to 10
• Three face cards: Jack, Queen, and King
• An Ace

The composition of a standard deck is fundamental in probability problems concerning cards, as it helps determine the number of total and favorable outcomes, basic components of probability calculations.

Favorable Outcomes

In probability exercises, defining what is considered a favorable outcome is crucial. A favorable outcome is any result that meets the condition described in the problem. For the card problem given, favorable outcomes are those that result in selecting a face card. Face cards in a standard deck include:

• Jack
• Queen
• King

Since each suit (hearts, diamonds, clubs, spades) contains one of each face card, there are 3 face cards per suit.
With 4 suits, this gives a total of 12 favorable outcomes. Identifying these favorable outcomes allows us to calculate probabilities accurately by comparing them against the total number of possible outcomes.

Simplifying Fractions

Simplifying fractions is an important step in solving probability problems, allowing for easier interpretation of results. To simplify a fraction means finding its lowest terms. This process involves dividing both the numerator and denominator by their greatest common divisor (GCD).Consider the probability of drawing a face card from a deck of cards, expressed as the fraction \( \frac{12}{52} \). To simplify:

• Find the GCD of 12 and 52, which is 4.
• Divide both numerator and denominator by 4: \( \frac{12 \div 4}{52 \div 4} \).
• Result: \( \frac{3}{13} \).

This simplification provides a clearer view of the probability, helping in further analysis or comparison of different probabilities.