Problem 27 - 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 two

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We're asked to find the probability of drawing a card that is a 'two' from a standard deck. A standard deck has 52 cards, consisting of 4 suits (hearts, diamonds, clubs, spades), each with 13 cards (Ace through King).

Step 2: Identify the Favorable Outcomes

Count how many 'two' cards are in a deck. Since each suit contains one 'two', there are 4 'twos' in a full deck of cards.

Step 3: Determine the Total Outcomes

Recall that the total number of possible outcomes when drawing one card from the deck is 52, because there are 52 cards in the deck.

Step 4: Calculate the Probability

The probability of an event is given by the formula \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). Thus, the probability of drawing a 'two' is \( \frac{4}{52} \).

Step 5: Simplify the Fraction

Simplify the fraction \( \frac{4}{52} \) by dividing both the numerator and the denominator by 4, resulting in \( \frac{1}{13} \).

Key Concepts

Standard Deck of Cards

A standard deck of cards is an essential concept in probability and card games. It is composed of 52 cards, which are evenly distributed across four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, starting with Ace and ending with King. The cards in each suit are: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King.
Understanding the basic composition of a deck helps in calculating probabilities, as various probability problems often involve drawing one or more cards. In this specific exercise, the focus is on drawing a particular card from the deck, such as a 'two'. To solve this, it is crucial to know how many total cards and specific cards of interest are available in the deck.

Favorable Outcomes

In probability, when we talk about favorable outcomes, we refer to the number of outcomes that meet the criteria of the event we are interested in. In this exercise, we are concerned with the event of drawing a 'two' from the deck.
To find favorable outcomes, count how many variations of 'two' are present in a standard deck. Since each suit contains one 'two', with suits being hearts, diamonds, clubs, and spades, there are four 'twos' total. Thus, the number of favorable outcomes when drawing a 'two' is 4. This counting becomes the numerator when calculating probability.

Total Outcomes

The term total outcomes in probability refers to the total number of possible results in a given situation. When discussing a standard deck of cards, total outcomes mean all the possible cards you can draw.
Since there are 52 cards in a complete deck, the total number of outcomes is 52. This figure serves as the denominator when determining the probability of drawing a specific card, such as the 'two'. Understanding the total outcomes is critical in forming the base against which favorable outcomes are compared.

Simplifying Fractions

Simplifying fractions is a common task in mathematics, particularly in probability calculations. It involves reducing a fraction to its simplest form, which can make understanding and communicating probability results easier.
The initial probability of drawing a 'two' from our deck was expressed as a fraction: \( \frac{4}{52} \). To simplify, divide both the numerator (number of favorable outcomes) and the denominator (total outcomes) by their greatest common divisor, which in this case is 4.

- Divide 4 by 4, which equals 1.- Divide 52 by 4, which equals 13.Thus, the simplified probability of drawing a 'two' becomes \( \frac{1}{13} \). Simplifying keeps calculations neat and is a skill used frequently in probability discussions.