Problem 23 - Precalculus with Limits

Question

In Exercises 21 - 24, find the probability for the experiment of selecting one card from a standard deck of \( 52 \) playing cards. The card is a red face card.

Step-by-Step Solution

Answer

The probability of drawing a red face card from a standard deck is \( \frac{3}{26} \).

Step 1: Identify Total Outcomes

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

Step 2: Identify Favorable Outcomes

A red face card can be either a Heart or a Diamond, and since there are three face cards (King, Queen, Jack) in each suit, and there are 2 red suits, there are \( 2 * 3 = 6 \) red face cards.

Step 3: Calculate Probability

The probability of a specific outcome can be determined by dividing the number of favorable outcomes by the total number of outcomes. In this case, that would be \( \frac{6}{52} = \frac{3}{26} \).

Key Concepts

Deck of Cards

When we talk about a 'deck of cards,' we generally mean a standard deck used in a typical card game. This deck has 52 unique cards. All these cards are divided evenly into four suits:

• Hearts
• Diamonds
• Clubs
• Spades

A standard deck has non-numeric cards known as 'face cards.' Face cards include the King, Queen, and Jack. Each suit contains exactly three face cards. Understanding how a deck is structured helps in calculating probabilities for any card-related experiments.

Favorable Outcomes

'Favorable outcomes' refer to the outcomes that we are interested in when performing a probability experiment. In the context of drawing a card from a deck, it means drawing a card that meets the specific criteria we are looking for. For instance, in our given exercise, we are interested in the red face cards.
Red cards only belong to two suits: Hearts and Diamonds. Each of these suits has exactly three face cards: King, Queen, and Jack. This gives us a total of 6 red face cards. These 6 cards are the 'favorable outcomes' in this scenario.

Probability Calculation

Calculating probability involves understanding both the favorable outcomes and all possible outcomes. Probability is the measure of how likely an event is to occur. It is calculated using the formula:\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]In our card problem, the total number of outcomes is 52 (the total number of cards), and the number of favorable outcomes is 6 (red face cards). Plugging these numbers into our formula gives:\[ \text{Probability} = \frac{6}{52} = \frac{3}{26} \]This fraction simplifies the probability of drawing a red face card from a standard deck. It's always important to simplify fractions in probability to give the simplest form.

Standard Deck

A standard deck of cards is an essential component in probability problems involving cards. It consists of 52 cards divided into four suits, with each suit having 13 cards. These suits are divided into two colors: red (Hearts and Diamonds) and black (Clubs and Spades).

Within each suit, there are numbered cards from 2 through 10, and three face cards: the Jack, Queen, and King. Ace can be either considered high or low, depending on the game.

• Face cards: King, Queen, Jack
• Total face cards in a deck = 3 (in each suit) x 4 suits = 12
• Red face cards = 3 (from Hearts) + 3 (from Diamonds) = 6

Understanding the structure of a standard deck is pivotal when tackling probability questions related to card games.