Problem 18 - Algebra and Trigonometry

Question

You are dealt one card from a standard 52-card deck. Find the probability of being dealt $$\text{a diamond.}$$

Step-by-Step Solution

Answer

The probability of drawing a diamond from a standard 52-card deck is 0.25 or 25%.

Step 1: Identify Total Number of Outcomes

In a standard deck, there are 52 cards. So, the total number of outcomes is 52.

Step 2: Identify Number of Favorable Outcomes

There are 13 diamonds in a deck. So, the number of favorable outcomes is 13.

Step 3: Calculate Probability

The probability is calculated as the number of favorable outcomes divided by the total number of outcomes, in this case, 13 divided by 52. This simplifies to \( \frac{1}{4} = 0.25 \).

Key Concepts

Understanding Playing Cards

A standard deck of playing cards is universally understood, but let's explore its structure. A full deck comprises 52 cards. These cards are equally divided across four suits:

• Hearts
• Diamonds
• Clubs
• Spades

Each suit contains thirteen cards, numbered from Ace through 10, and also includes the face cards: Jack, Queen, and King.

The diamond suit is one of the two red suits in the pack, alongside hearts. This structured, consistent setup allows us to perform probability calculations by recognizing patterns and ratios within the deck.

Exploring Probability Calculation

Probability is a fascinating concept which allows us to determine how likely an event is to happen. To calculate probability, you use the formula:\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]In our card example, we want to find out the likelihood of drawing a diamond from a 52-card deck.

Knowing the structure of the deck helps. With 13 diamonds among 52 cards, the calculation becomes straightforward:

• Number of favorable outcomes: 13 (since there are 13 diamonds)
• Total number of possible outcomes: 52 (as there are 52 cards in total)

Plugging these numbers into our formula gives us:\[ \text{Probability} = \frac{13}{52} = \frac{1}{4} \] This means there is a 0.25 or 25% chance of drawing a diamond.

Identifying Favorable Outcomes

The concept of favorable outcomes is crucial in probability theory. They are the specific outcomes from a probability space that result in a successful or desired event. In our card scenario, drawing a diamond is the desired event. Favorable outcomes are what you count to determine the numerator in your probability formula.

Given the deck structure:

• There are 13 total diamond cards, meaning 13 favorable outcomes.

Understanding this helps simplify probability problems because it focuses on isolating only those outcomes that lead to success.

By identifying these outcomes and comparing them against all possible outcomes, you can reliably calculate probability, improving your understanding and intuition about how likely an event is.