Problem 27 - College Algebra

Question

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

Step-by-Step Solution

Answer

The probability of drawing a 'Two' is \( \frac{1}{13} \).

Step 1: Understand the Deck of Cards

A standard deck of cards contains 52 cards. These are divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, and each suit has one card of each rank, including a 'Two'.

Step 2: Identify the Favorable Outcomes

In a deck, there is one 'Two' card in each of the four suits. Thus, there are 4 'Two' cards in total: Two of Hearts, Two of Diamonds, Two of Clubs, and Two of Spades.

Step 3: Calculate the Probability

The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Here, the number of favorable outcomes (drawing a 'Two') is 4, and the total number of possible outcomes (drawing a card) is 52. Therefore, the probability is given by \( \frac{4}{52} \).

Step 4: Simplify the Fraction

To simplify \( \frac{4}{52} \), divide both the numerator and the denominator by the greatest common divisor, which is 4. This simplifies to \( \frac{1}{13} \).

Key Concepts

Deck of Cards

A standard deck of cards is a key element in understanding many probability exercises. It consists of 52 individual cards evenly distributed across four distinct suits:

• Hearts
• Diamonds
• Clubs
• Spades

Each suit holds 13 cards, ranked from Ace to King. This includes ranks such as numbers, from 'Two' to 'Ten', and face cards such as 'Jack', 'Queen', and 'King'. Keeping track of this structure helps in identifying various probabilities simply by recognizing the uniformity and distribution across the suits. The consistent layout of suits and their compositions assists in predicting outcomes and solving probability problems efficiently.

Favorable Outcomes

In probability, the term "favorable outcomes" refers to the specific outcomes that meet the criteria of the event we are investigating. For instance, if we are trying to determine the probability of drawing a 'Two' from a deck of cards, we need to identify how many 'Two' cards exist in the deck.

• In a standard 52-card deck, there is exactly one 'Two' of each suit.
• This totals up to four 'Two' cards: Two of Hearts, Two of Diamonds, Two of Clubs, and Two of Spades.

This set of four cards represent our favorable outcomes. Calculating probability involves determining the ratio of these favorable outcomes to the total number of possible outcomes, which in this case is the total number of cards in the deck. This helps students visualize the exact elements contributing to the probability of the event taking place.

Simplifying Fractions

Simplifying fractions is a fundamental concept not only in probability but also in general mathematics. The objective is to express a fraction in its simplest form where the numerator and denominator are as small as possible. To simplify a fraction, you follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

For instance, when you calculate the probability of drawing a 'Two' as \(\frac{4}{52}\), you simplify it by finding the GCD of 4 and 52, which is 4.

Divide both 4 and 52 by 4 to get \(\frac{1}{13}\). This fraction represents the simplest form of the original probability, making it easier to understand and communicate.

Simplifying fractions helps students better comprehend probabilities by presenting results in a more straightforward manner.