Problem 30 - College Algebra

Question

Involve a deck of 52 cards. If you are dealt 3 cards from a shuffled deck of 52 cards, find the probability that all 3 cards are picture cards.

Step-by-Step Solution

Answer

The probability that all 3 cards are picture cards when dealt 3 cards from a shuffled deck of 52 cards is \( P = \frac{C_{success}}{C_{total}} \)

Step 1: Calculate Total Number of Potential Outcomes

The total number of outcomes is calculated using combination as there are 52 cards and we draw 3, hence \( C_{total} = C(52,3) = \frac{52!}{3!(52-3)!} \).

Step 2: Calculate Number of Successful Outcomes

The 'successful' outcomes we want are the ways to pick 3 picture cards out of the 12 picture cards in the deck. Hence the number of successful outcomes is \( C_{success} = C(12,3) = \frac{12!}{3!(12-3)!}\) .

Step 3: Calculate the Probability

The probability is the number of successful outcomes divided by the total number of outcomes. Hence, \( P = \frac{C_{success}}{C_{total}} \) .

Key Concepts

Combinatorics

Combinatorics is an essential branch of mathematics that deals with counting, arrangement, and combination of objects. It's all about finding the possible ways things can happen. In our everyday life, combinatorics is useful in a wide range of scenarios, from planning and scheduling to understanding complex systems. For instance, it helps us find out how many different ways we can choose a selection of items or how ways we can sort these items. Combinatorics is the foundation of various probability problems, where the goal is to count possible outcomes.
In the context of card games and other similar exercises, combinatorics helps us determine the number of possible ways to draw a certain number of cards from a larger set, like a standard deck of 52 cards. These calculations are crucial for understanding probabilities, such as the likelihood of drawing specific card combinations.

Card Probabilities

Card probabilities are the likelihood of certain outcomes when drawing cards from a deck. In card games, understanding these probabilities helps players make informed decisions and strategize their next moves. Probability is calculated as the ratio of favorable outcomes to the total number of possible outcomes.
Consider the situation where we need the probability of drawing specific cards, such as picture cards from a deck. A standard deck has 52 cards, out of which 12 are picture cards: four Jacks, four Queens, and four Kings. When you draw 3 cards from this deck, card probabilities will help you calculate how likely it is for all three to be picture cards. By understanding these probabilities, players can enhance their strategies and improve their gameplay.

Permutations and Combinations

Permutations and combinations are key concepts within combinatorics that help us figure out different arrangements and selections of objects. They are different but related tools.

• Permutations: Focused on arranging objects where the order matters. For example, if you want to know how many different ways you can arrange 3 of those picture cards you've drawn, you'd be dealing with permutations.
• Combinations: These are used when the order doesn't matter, just the selection. This is what we use in our card probability problem. We use combinations to calculate how many ways we can choose 3 picture cards out of the 12 available picture cards. It is expressed mathematically as \( C(n, r) = \frac{n!}{r!(n-r)!} \) where \( n \) represents the total number of items to choose from, and \( r \) is the number of items to choose.

Understanding these differences is crucial in correctly solving probability problems involving arranging or choosing items from a larger set.

Deck of Cards

A standard deck of cards is a familiar tool used worldwide for various games and exercises. This deck consists of 52 cards which are divided equally into 4 suits: Hearts, Diamonds, Clubs, and Spades. Each suit contains 13 cards that are ranked from Ace through 10, followed by three picture cards: Jack, Queen, and King.
This structure of a deck means whenever you're considering probabilities or combinations involving cards, you're also considering this breakdown and distribution of cards. Whether you're dealing with games or statistical exercises, understanding the makeup of a deck is the starting point to solving complex card-related problems.