Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, one card is drawn from a standard deck of 52 cards. Find the probability of drawing the following: A non-ace

Step-by-Step Solution

Answer

The probability of drawing a non-ace card is \( \frac{12}{13} \).

Step 1: Understand the Question

We need to find the probability of drawing a non-ace card from a standard deck of 52 cards. A standard deck has 4 suits, each with 13 cards. Among them, there are 4 aces (one in each suit).

Step 2: Calculate Total Non-Ace Cards

Subtract the number of aces in a deck from the total number of cards. There are 52 cards in total and 4 aces, so the number of non-ace cards is 52 - 4 = 48.

Step 3: Calculate the Probability

The probability of drawing a non-ace card is the ratio of non-ace cards to the total number of cards. This is given by the formula \( \frac{\text{Number of non-ace cards}}{\text{Total number of cards}} \). Thus, the probability is \( \frac{48}{52} \).

Step 4: Simplify the Fraction

To simplify \( \frac{48}{52} \), find the greatest common divisor of 48 and 52, which is 4. Dividing both the numerator and the denominator by 4 gives \( \frac{12}{13} \). Therefore, the probability of drawing a non-ace card is \( \frac{12}{13} \).

Key Concepts

Standard Deck of Cards

A standard deck of cards is like a small universe of possibilities. Imagine a closed system of 52 well-organized cards. It consists of four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, adding up to the complete 52 cards.
Within each suit, you have a sequence of ranks from Ace through Kings.

• Ace
• Numbers 2 through 10
• Jack, Queen, King

This consistency across suits makes card games and probability calculations very structured.
Knowing how a deck is organized helps you navigate problems about card probability with ease. As you delve deeper, you'll see how understanding this structure becomes a key factor in calculating probabilities.

Non-Ace Card

When we refer to non-ace cards, we are excluding all Aces from our calculations. In a standard deck of 52 cards, there are 4 Aces, one in each suit. To find the number of non-ace cards, you subtract the 4 aces from the total number:
52 total cards - 4 aces = 48 non-ace cards.
This category of cards includes every card that isn't an Ace, which is important for probability calculations.

• Calculate total number of non-ace cards: 48
• Know that non-ace cards span numbers 2 through 10, and the face cards (Jack, Queen, and King)
  

Understanding which cards qualify as non-aces is crucial as it directly impacts the computation of probability every time you draw a card.

Simplifying Fractions

Simplifying fractions is a technique used to express a fraction in its simplest form by finding a common factor that both the numerator and the denominator share.
This technique makes the fraction easier to understand and present. When you calculate a probability, like drawing a non-ace card, you might get a fraction that needs simplification.
Start by finding the greatest common divisor (GCD) of both parts of the fraction.
In our case, for the fraction \( \frac{48}{52} \), the GCD is 4. You then divide both the numerator and the denominator by this number:

• \( 48 \div 4 = 12 \)
• \( 52 \div 4 = 13 \)

This renders the simplified fraction \( \frac{12}{13} \). Such simplification does not change the value of the fraction, but it does make the probability more digestible, which helps to better understand and communicate statistical concepts.