Problem 26 - Algebra for College Students

Question

Find the probability of each of the following events: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: Identify Total Outcomes

First, identify the total number of outcomes in the deck of cards. A standard deck has 52 cards, so the total number of possible outcomes is 52.

Step 2: Identify Favorable Outcomes

Next, identify the number of favorable outcomes for the event. There are 2 red jacks in a deck of cards (one from hearts and one from diamonds), so the number of favorable outcomes is 2.

Step 3: Calculate Probability

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Thus, the probability of drawing a red jack is given by:\[P(\text{{red jack}}) = \frac{\text{{Number of red jacks}}}{\text{{Total cards}}} = \frac{2}{52} = \frac{1}{26}\]

Key Concepts

Favorable Outcomes

When dealing with probability problems, understanding the concept of favorable outcomes is crucial. A favorable outcome refers to the specific result or set of results that satisfy the criteria of an event you are interested in. In simpler terms, these are the outcomes that lead to the success of the event or scenario you are analyzing.
For example, if you are drawing a card from a deck and you want to find the probability of drawing a red jack, the favorable outcomes are those where a red jack is drawn. In a standard deck, there are exactly two red jacks, the Jack of Hearts and the Jack of Diamonds.
Whenever you approach a probability problem, make sure to clearly define and count the favorable outcomes. This will significantly simplify the process of calculating the probability of the event.

Total Outcomes

The total outcomes in probability refer to the complete set of possible outcomes that can happen in a given scenario. When you determine total outcomes, you account for every single possibility without considering any restrictions from the event. Thus, it provides the baseline over which you calculate your probability.
In the context of a card-drawing problem, a standard deck of cards represents the universe of possible outcomes. Since a deck has 52 cards, there are 52 total outcomes when drawing one card. Each card is unique, so each draw provides a distinct outcome.
Thus, to calculate probabilities effectively, always identify the total number of outcomes first. This way, you have the basic framework required for computing how likely your favorable outcomes are.

Standard Deck of Cards

A standard deck of cards is a familiar tool used in probability problems. Understanding its structure is fundamental to handling such exercises smoothly.
A standard deck comprises 52 cards and is divided evenly into four suits: hearts, diamonds, clubs, and spades. Each suit consists of 13 cards, which include numbers 2 through 10, and the face cards: Jack, Queen, King, and Ace.

• Hearts and diamonds are red
• Clubs and spades are black

Knowing this structure allows you to quickly determine both total and favorable outcomes for various probability exercises. For instance, if you're asked about red cards or certain face cards, you can refer back to the composition of the deck to pinpoint the number you need.
Mastering the properties of a standard deck makes tackling probability problems more intuitive and less daunting.