Problem 26 - Algebra for College Students

Question

\text { A red jack is drawn. } \frac{1}{26}

Step-by-Step Solution

Answer

The probability of drawing a red jack is \( \frac{1}{26} \).

Step 1: Understand the Problem

We need to find the probability of drawing a red jack from a standard 52-card deck. In a standard deck, there are 26 red cards, with two of them being red jacks (one Jack of Hearts and one Jack of Diamonds).

Step 2: Calculate the Total Possible Outcomes

From a 52-card deck, the total number of possible outcomes (ways to draw a card) is 52.

Step 3: Calculate the Favorable Outcomes

The number of favorable outcomes is the number of red jacks in the deck. There are 2 red jacks in a standard deck.

Step 4: Apply the Probability Formula

The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Thus, the probability of drawing a red jack is calculated as follows: \[ \text{Probability of drawing a red jack} = \frac{2}{52} = \frac{1}{26} \]

Step 5: Simplify if Necessary

The fraction \( \frac{2}{52} \) is already simplified to \( \frac{1}{26} \). Hence, no further simplification is needed.

Key Concepts

standard deck of cards

A standard deck of cards consists of 52 cards. These cards are divided into four suits: Hearts, Diamonds, Clubs, and Spades. Each suit has 13 cards, which range from Ace to King. Each of these suits contains cards numbered 2 through 10, and face cards which are the Jack, Queen, and King.
Among these suits, Hearts and Diamonds are red, while Clubs and Spades are black. This division means there are 26 red cards and 26 black cards.

52 cards in total

4 suits: Hearts, Diamonds (red), Clubs, Spades (black)

Each suit has 13 cards: Ace through King

Knowing the composition of a standard deck is crucial for solving probability problems involving cards, as it lets you determine both the total possible outcomes and the favorable outcomes for any card drawing scenario.

favorable outcomes

In probability, a favorable outcome is any outcome in which the event we are interested in occurs. In the context of our exercise, our event of interest is drawing a red jack.
Out of a standard deck's 52 cards, there are two red jacks: the Jack of Hearts and the Jack of Diamonds.
Therefore, there are 2 favorable outcomes for drawing a red jack. Understanding favorable outcomes helps us quantify how likely an event is to happen, as they form the numerator in the probability calculation.

2 favorable outcomes: Jack of Hearts, Jack of Diamonds

Identifying these outcomes enables us to apply the probability formula directly and calculate the probability of drawing a card of interest.

card drawing probability

The probability of an event is a measure of the likelihood that the event will occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
For our exercise of drawing a red jack from a standard deck of cards, the total number of possible outcomes is 52, as you can draw any card from the deck.

Total possible outcomes: 52

Favorable outcomes (red jacks): 2

The probability formula is:\[\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}\]Plugging in our numbers:\[\text{Probability of drawing a red jack} = \frac{2}{52} = \frac{1}{26}\]This simplified fraction represents the probability of drawing a red jack from the deck.