Problem 27 - Algebra 2

Question

Two cards are drawn from a standard deck of cards. Find each probability. \(P(\text { both jacks or both face cards) }\)

Step-by-Step Solution

Answer

The probability is approximately 0.0498.

Step 1: Identify Total Possible Outcomes

A standard deck of cards consists of 52 cards. If two cards are drawn, the total number of possible combinations is calculated using the combinations formula: \( \binom{52}{2} \). Thus, the total possible outcomes are 1326.

Step 2: Calculate Probability Both Cards are Jacks

There are 4 Jack cards in the deck. To find the probability of drawing two Jacks, calculate the number of ways to choose 2 Jacks out of 4: \( \binom{4}{2} = 6 \). Divide by the total combinations: \( \frac{6}{1326} \approx 0.0045 \).

Step 3: Calculate Probability Both Cards are Face Cards

There are 12 face cards in a deck (Jacks, Queens, Kings in each suit). The number of ways to choose 2 face cards is \( \binom{12}{2} = 66 \). Divide by the total combinations to find the probability: \( \frac{66}{1326} \approx 0.0498 \).

Step 4: Calculate Combined Probability (once without Overlapping)

To find the probability of either drawing both Jacks or both face cards, use the previously calculated probabilities. However, note that 'both Jacks' are already counted in 'both face cards'. Thus, the probability is simply the same as the probability of drawing two face cards: \( 0.0498 \).

Key Concepts

Combinations Formula

When determining how many ways we can choose a certain number of items from a larger set, we use the combinations formula. This formula helps us find how many different ways we can draw two cards from a deck of 52 cards, for instance. The formula is written as \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

• \( \binom{52}{2} \): Here, we get the total number of ways to choose 2 cards from 52, resulting in 1326 possible combinations.
• \( \binom{4}{2} \): For finding how we can choose 2 Jacks out of the 4 available in the deck.
• \( \binom{12}{2} \): Used for finding the number of ways to pick 2 face cards from the 12 in the deck.

This approach reduces the complexity of counting outcomes manually and ensures accuracy.

Standard Deck of Cards

A standard deck of cards is a familiar tool in probability questions. Understanding its structure is critical in these problems. It contains 52 cards divided into 4 suits: Hearts, Diamonds, Clubs, and Spades. Each suit consists of 13 cards, ranging from Ace to King. Within these 13:

• Numbered cards from 2 to 10.
• Face cards: Jack, Queen, and King.

The uniform structure of the deck simplifies calculations, like determining the likelihood of drawing a card of a particular type.

Face Cards Probability

Face cards in a deck are the Jacks, Queens, and Kings. Each of these appears four times, once in each suit, totaling 12 face cards altogether. To find the probability of drawing two face cards:1. Use the combinations formula to select 2 cards from 12 face cards: \( \binom{12}{2} = 66 \).2. Divide by the total combinations of drawing any two cards: \( \frac{66}{1326} \approx 0.0498 \).This probability reflects how likely it is to draw two face cards in one draw.

Jack Cards Probability

Jacks are one subset of face cards, with each deck containing 4 Jacks — one per suit. To compute how likely it is to draw two Jack cards:1. Calculate the number of ways to select 2 Jacks out of 4: \( \binom{4}{2} = 6 \).2. Then, find the proportion of these specific Jacks selections to the total combinations: \( \frac{6}{1326} \approx 0.0045 \).The concept helps clarify the specific odds of drawing pairs of certain types of cards.