Problem 34 - College Algebra

Question

You randomly select one card from a 52-card deck. Find the probability of selecting: a 7 or an 8.

Step-by-Step Solution

Answer

The probability of drawing a 7 or an 8 from a deck of 52 cards is \( \frac{8}{52}\) or approximately \(.154\), when rounded to three decimal places.

Step 1: Determine Total Number of Outcomes

The total number of outcomes is the total number of cards in the deck, which is \(52\).

Step 2: Determine Favorable Outcomes

Now the task is asking for selecting a 7 or an 8, meaning any card that is either of these two counts as a success. There are 4 sevens and 4 eights, adding to \(4 + 4 = 8\) successful outcomes.

Step 3: Calculate the Probability

The probability of an event can be found by dividing successful outcomes by total outcomes. So in this case, the probability is \( \frac{8}{52}\).

Key Concepts

Card Decks

A standard card deck is an essential tool in probability theory, often used to illustrate basic probability concepts. Each deck is comprised of 52 cards, divided into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards, ranging from Ace through to King. These components make the card deck ideal for probability exercises as it provides both a finite set of total outcomes and distinctive individual outcomes that can be easily categorized.

• Hearts and diamonds are red suits.
• Clubs and spades are black suits.
• Each suit includes face cards - Jack, Queen, and King.

Understanding the structure of a card deck helps in identifying the total number of possible outcomes in a probability problem, which is often the first step when calculating the probability of a given event.

Favorable Outcomes

Favorable outcomes refer to the specific outcomes of an event that satisfy the conditions you're interested in. In probability, defining what exactly you're looking for is crucial. In our exercise, we want to find the probability of selecting a 7 or an 8 from a standard 52-card deck. Since each of these numbers appears once in each of the four suits, we have:

• 4 sevens (one in each suit).
• 4 eights (one in each suit).

Thus, there are a total of 8 favorable outcomes when drawing a card that is either a 7 or an 8. This step often involves counting or listing outcomes to ensure accuracy.

Total Outcomes

To calculate the probability of an event, determining the number of total outcomes is fundamental. In any complete set, like a deck of cards, total outcomes refer to the entire collection of possibilities. A standard deck has 52 cards, this number represents every potential outcome when a single card is drawn. Understanding this helps to set the basis for further calculations in probability. The total number of outcomes is foundational because it acts as the denominator in probability calculations. By comparing the number of favorable outcomes to the total number of outcomes, we gain insight into the likelihood of a particular event occurring.

Event Probability

Event probability quantifies the likelihood of a specific event happening. It is calculated by dividing the number of favorable outcomes by the total number of outcomes. For example, to find the probability of drawing a 7 or an 8, we divide the number of favorable outcomes (8) by the total number of outcomes (52). The formula is:\[P( ext{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}\]So, the probability is:\[P( ext{7 or 8}) = \frac{8}{52} = \frac{2}{13}\]This shows that there is a 2 in 13 chance that the next card drawn will be a 7 or an 8. Probability offers a way to express this chance numerically, facilitating understanding and decision-making.