Problem 35 - College Algebra

Question

You randomly select one card from a 52-card deck. Find the probability of selecting: a red 2 or a black 3

Step-by-Step Solution

Answer

The probability of selecting either a red 2 or a black 3 from a 52-card deck is \(\frac{1}{13}\).

Step 1: Determining the number of desired cards

In a 52-card deck, there are two red 2s (diamonds and hearts), and two black 3s (clubs and spades). Thus, the total number of desired cards is 2 + 2 which equals 4.

Step 2: Calculate total number of outcomes

The total number of outcomes is equivalent to the total number of cards in the deck which is 52.

Step 3: Calculating Probability

The probability is computed by dividing the number of desired outcomes by the total number of outcomes. Thus, the probability of selecting either a red 2 or a black 3 is \(\frac{4}{52} = \frac{1}{13}\)

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and combination of elements within a set. It helps us calculate probabilities by allowing us to figure out how many ways we can arrange or select items from a set.
For example, in our card problem, combinatorics helps determine the number of specific cards we can draw – such as the two red 2s and the two black 3s from a deck of 52 cards.

• When we say there are two red 2s, we count the 2 of hearts and the 2 of diamonds.
• For the black 3s, we count the 3 of clubs and the 3 of spades.

Counting these specific cards gives us 4 cards of interest out of a deck of 52. Combinatorics provides the foundational tools to make these precise calculations, ensuring we consider only the configurations relevant to our problem. This methodical counting process is the first step in computing probabilities effectively.

Card Probability

Calculating card probabilities revolves around determining the likelihood of drawing a specific card or set of cards from a standard 52-card deck. Each card in the deck provides different outcomes, and understanding these is key to solving probability problems.
In our problem, we want to determine the probability of drawing either a red 2 or a black 3. To find this:

• We identified 4 favorable outcomes: two red 2s and two black 3s.
• The total number of possible outcomes is the total number of cards, which is 52.

Then by dividing the number of favorable outcomes (4) by the total number of possible outcomes (52), we validate that the probability of drawing a red 2 or a black 3 from the deck is \(\frac{1}{13}\). Card probability leverages both the total size of the deck and the specific characteristics of the cards to yield an accurate measure of likelihood.

Basic Probability Concepts

To understand basic probability concepts, it's important to grasp where probabilities originate and how they are calculated. Probability measures how likely an event is to occur and is calculated using a simple formula: the number of desired outcomes divided by the total number of possible outcomes.

• Desired Outcomes: These are the events or results we're interested in – in our card exercise, drawing a red 2 or a black 3.
• Total Outcomes: This is the complete set of all possible options, here represented by all 52 cards in a deck.

This results in a probability expressed as a fraction or a decimal between 0 and 1 – where 0 means the event is impossible, and 1 means the event is certain. In our card example, the probability of drawing a red 2 or a black 3 is \(\frac{1}{13}\), indicating it's a possible yet not very likely event. Understanding these basics helps in virtually any scenario where determining likelihood is required, making this concept invaluable across countless applications.