Problem 29 - Algebra 2

Question

Two cards are drawn from a standard deck of cards. Find each probability. \(P(\text { both either black or an ace) }\)

Step-by-Step Solution

Answer

The probability is \( \frac{63}{221} \).

Step 1: Understand the Problem

We are asked to find the probability of drawing two cards from a deck such that both are either black or an Ace. The deck has 52 cards with 26 black cards (13 clubs and 13 spades) and 2 Aces which are Red (one heart and one diamond), as the 2 black Aces are already accounted for in the black cards.

Step 2: Determine Total Outcomes

In a standard deck, there are 52 cards. The number of ways to draw 2 cards from this deck can be calculated using combinations: \[ \binom{52}{2} = \frac{52 \times 51}{2} = 1326. \] This is the total number of outcomes.

Step 3: Determine Favorable Outcomes

The favorable outcomes are those where both cards drawn are either black or an Ace.- There are 28 favorable cards (26 black cards + 2 additional Aces).Now, calculate the number of ways to draw 2 cards from these 28:\[ \binom{28}{2} = \frac{28 \times 27}{2} = 378. \] These are the favorable outcomes.

Step 4: Calculate the Probability

The probability of drawing two cards such that both are either black or an Ace is given by the ratio of favorable outcomes to total outcomes:\[ P(\text{both cards black or an Ace}) = \frac{378}{1326}. \] Simplifying this fraction, we have:\[ P(\text{both cards black or an Ace}) = \frac{63}{221}. \]

Key Concepts

Combinatorics

Combinatorics is a fascinating field of mathematics that involves counting, arranging, and combining items in specific ways. When tackling probability problems, combinatorics is essential because it helps calculate the number of ways events can occur.
Combinatorics involves concepts like permutations and combinations. For problems involving selection, such as card drawing, combinations are most useful. Combinations are used when the order of selection doesn't matter. For example, choosing 2 cards from a deck of 52.
Understanding the difference between permutations (where order matters) and combinations (where order doesn't matter) is crucial in solving probability problems efficiently.

Deck of Cards

A standard deck of cards is a common tool used in probability exercises. It consists of 52 cards divided into four suits: spades, clubs, hearts, and diamonds. Each suit contains 13 cards ranging from two to ten, plus a jack, queen, king, and ace.
In problems involving card draws, it's important to note that half the cards are black (spades and clubs), and the other half are red (hearts and diamonds). Knowing the composition of a deck assists in identifying which cards are relevant to a specific problem, such as drawing a black card or an ace.

• 26 black cards: 13 spades + 13 clubs.
• 2 black aces count towards black cards.
• Aces are special as there are 4 in a deck, one in each suit.

Understanding these basics helps in accurately setting up the conditions for calculating probabilities.

Favorable Outcomes

Favorable outcomes are the number of successful ways an event can happen based on a certain condition, such as drawing cards that are either black or aces.
For the exercise, identify how many cards meet the criteria: either being a black card or an ace.- Here, there are 28 favorable cards: 26 black cards and 2 additional red aces.
Once identified, the next step is using combinations to determine how many ways you can draw two such cards from the 28 available.
This is calculated using the combination formula \( \binom{n}{r} \), where \( n \) is the total number of available favorable outcomes and \( r \) is the number of draws. For this problem, it is \( \binom{28}{2} \).

Total Outcomes

Total outcomes refer to the complete set of possible results for an action. In probability, it's crucial to identify this number as it forms the denominator when calculating the probability of an event.
When drawing cards from a deck, the total outcomes are determined by the number of possible card combinations. Here, we're looking at drawing 2 cards from a full deck of 52.
Using the combination formula \( \binom{n}{r} \), where \( n \) is 52 (the total number of cards), and \( r \) is the number of draws (2 cards in this case), we compute:

• \( \binom{52}{2} = \frac{52 \times 51}{2} = 1326. \)

This total number of outcomes is essential for calculating the probability, where the ratio of favorable outcomes to total outcomes gives the chance of drawing two black cards or aces.