Problem 43 - Algebra 2

Question

Find each probability if 13 cards are drawn from a standard deck of cards and no replacement occurs. \(P(\text { all red cards })\)

Step-by-Step Solution

Answer

The probability is approximately 0.000016.

Step 1: Understand the Problem

We need to calculate the probability of drawing 13 red cards from a deck of 52 cards without replacement. In a deck, there are 26 red cards (hearts and diamonds) and 26 black cards (spades and clubs).

Step 2: Assign Values

The number of favorable outcomes, i.e., selecting 13 red cards from 26, can be represented by the combination \( \binom{26}{13} \). The total number of ways to select 13 cards from the entire deck is \( \binom{52}{13} \).

Step 3: Calculate the Total Combinations

Compute the total number of combinations to draw any 13 cards from the deck: \( \binom{52}{13} = \frac{52!}{13!(52-13)!} \).

Step 4: Calculate Favorable Combinations

Compute the combinations for drawing 13 red cards from 26 red cards: \( \binom{26}{13} = \frac{26!}{13!(26-13)!} \).

Step 5: Calculate Probability

The probability is obtained by dividing the number of favorable combinations by the total combinations: \( P(\text{all red cards}) = \frac{\binom{26}{13}}{\binom{52}{13}} \).

Step 6: Compute Numerical Values

Calculate the numerical values: \( \binom{26}{13} = 10400600 \) and \( \binom{52}{13} = 635013559600 \). Use these values to find the probability: \( P(\text{all red cards}) = \frac{10400600}{635013559600} \approx 0.000016 \).

Key Concepts

Combinations

Combinations are a mathematical way to determine how many ways a particular group of items can be selected from a larger group. This is useful when the order of selection does not matter. In probability, combinations are essential for calculating events where the order is irrelevant. For example, when drawing cards, it doesn't matter in which order the cards are drawn, making combinations the ideal tool for calculation.

We can represent combinations using the binomial coefficient, denoted as \( \binom{n}{r} \). It reads as "n choose r" and is calculated by the formula:

• \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Here, \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial, which means the product of all positive integers up to that number.

Understanding combinations is crucial for solving problems like determining the probability of drawing a specific set of cards from a deck.

Standard Deck of Cards

A standard deck of cards consists of 52 cards, which are divided into four suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards: numbers 2 through 10, and the face cards - jack, queen, and king, along with the ace.

It's vital to understand the composition of a deck to perform probability calculations accurately. There are two red suits (hearts and diamonds) and two black suits (clubs and spades), giving a total of 26 red cards and 26 black cards.

Knowing how a deck is structured helps in problems like finding the probability of a specific color, suit, or sequence when drawing cards. For instance, if you want to calculate the probability of drawing all red cards, you must know there are initially only 26 red cards to choose from.

Card Probability Calculations

Calculating probability in card games involves determining the likelihood of a specific outcome occurring based on the composition of the deck. Probability is defined as the number of favorable outcomes divided by the total number of possible outcomes. When drawing cards, it's crucial to consider whether the draws are with or without replacement because it affects the total outcomes.

For example, in calculating the probability of drawing 13 red cards from a deck, we use combinations to find both the favorable and total outcomes.

• Favorable outcomes: the number of ways to draw 13 red cards from 26, \( \binom{26}{13} \).
• Total outcomes: the number of ways to draw any 13 cards from 52, \( \binom{52}{13} \).

The probability is then calculated by dividing the number of favorable outcomes by the total outcomes, expressed as:

• \[ P(\text{all red cards}) = \frac{\binom{26}{13}}{\binom{52}{13}} \]

Understanding this process is essential for tackling probability calculations in any context involving randomness and known possible outcomes, such as shuffling and drawing from a deck of cards.