Problem 10 - Algebra 2

Question

Two cards are drawn from a standard deck of cards. Find each probability if no replacement occurs. \(P(\text { two hearts })\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{17}\).

Step 1: Understanding the Problem

First, identify what is being asked. We need to find the probability of drawing two hearts consecutively from a standard deck of 52 cards without replacement.

Step 2: Calculating the Total Possible Outcomes

In a deck of 52 cards, when drawing two cards consecutively without replacement, the total combinations of drawing two cards can be calculated by choosing 2 cards from 52, which is 52 * 51.

Step 3: Counting Favorable Outcomes for the First Draw

There are 13 hearts in a deck of 52 cards. When drawing the first card, there are 13 favorable outcomes since any of the 13 hearts can be drawn.

Step 4: Counting Favorable Outcomes for the Second Draw

If one heart is drawn first, there are now 12 hearts left in a deck of 51 cards. Thus, there are 12 favorable outcomes for drawing a heart as the second card.

Step 5: Calculating the Probability

The probability of both events happening is the product of their individual probabilities: First card being a heart: \(\frac{13}{52}\). Second card being a heart after the first heart: \(\frac{12}{51}\). Therefore, the probability of both being hearts is: \(\frac{13}{52} \times \frac{12}{51}\). Simplifying this, we get \(\frac{1}{17}\).

Step 6: Finalizing the Probability

Multiply and simplify the fraction \(\frac{13}{52} \times \frac{12}{51}\) which results in \(\frac{156}{2652}\) and simplifies further to \(\frac{1}{17}\).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of objects. In the context of our problem, we are interested in how to calculate the number of ways two cards can be drawn from a deck of 52 cards without replacement. In combinatorics, when drawing without replacement, each draw affects the subsequent draws because the pool of possible outcomes shrinks.

For example:

• On the first draw, you have 52 potential cards to choose from.
• On the second draw, only 51 cards remain since one card has already been selected.

The total number of combinations of these draws is calculated by multiplying the number of choices for the first and the second draw, i.e., 52 * 51 in this problem. This simple multiplication only considers the order of draws, which is relevant since it impacts conditional probabilities.

Conditional Probability

Conditional probability is a concept used to determine the likelihood of an event, given that another event has already occurred. In our exercise, we want to know the probability of drawing two hearts in succession without replacing the first card.

To explore this:

• The first event is drawing a heart, with a probability of 13 out of 52 as hearts comprise 13 of the 52 cards in the deck.
• After one heart has been drawn, only 12 hearts out of the remaining 51 cards are left for the second draw.

Thus, the probability of drawing a second heart, given that the first card drawn was a heart, is 12/51. This probability relies on the condition that the first card drawn was a heart, illustrating conditional probability in action. Adjusting the sample space after each event is fundamental in such calculations.

Deck of Cards

A standard deck of cards consists of 52 unique cards, which provide a rich setting for probability exercises. This deck is traditionally divided into four suits: hearts, diamonds, clubs, and spades, each containing 13 cards. Understanding this structure is vital when determining probabilities because it helps us comprehend how probabilities unfold.

Each suit:

• Has an equal number of cards, affecting the calculation of drawing cards of a particular suit.
• Consists of cards numbered 2 through 10, and the face cards Jack, Queen, King, and Ace.

Whenever cards are drawn from this deck, knowing how many cards belong to each category helps in assessing the chances of specific outcomes, such as drawing cards of the same suit or obtaining specific ranks. Recognizing this composition facilitates understanding of probabilities like those dealt in the exercise, where calculating the probability of drawing two hearts hinges on knowing how many hearts are initially available and how the numbers change with each draw.