What is the probability of getting 2 hearts in a deck of cards

Understanding the Basics

To calculate the probability, we need to understand the basic properties of a deck of cards. A standard deck consists of 52 cards, with 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards.

Calculating the Probability

We are looking for the probability of drawing 2 hearts sequentially without replacement.

Step-by-Step Breakdown

1. Total Possible Outcomes: The total number of ways to draw 2 cards from a deck of 52 is given by the combination formula:

   $$\binom{52}{2} = \frac{52!}{2!(52-2)!} = \frac{52 \times 51}{2 \times 1} = 1326$$

2. Favorable Outcomes: The number of ways to draw 2 hearts out of 13:

   $$\binom{13}{2} = \frac{13!}{2!(13-2)!} = \frac{13 \times 12}{2 \times 1} = 78$$

3. Calculate the Probability:

   The probability $P$ of drawing 2 hearts in a row is the ratio of the number of favorable outcomes to the total possible outcomes:

   $$P(\text{2 hearts}) = \frac{\binom{13}{2}}{\binom{52}{2}} = \frac{78}{1326} = \frac{6}{102} = \frac{1}{17}$$

Final Result

Therefore, the probability of getting 2 hearts in a deck of cards is:

Explanation of the Principle of Combinations

Combinations refer to the selection of items from a larger pool, where the order of selection does not matter. This is a fundamental concept in combinatorics, a branch of mathematics concerned with counting, arrangement, and combination.

Key Concepts

1. Definition:

   • A combination is a way of selecting items from a group, such that the order of selection does not matter.
   • For example, selecting 3 fruits out of 5 different fruits, the combination {apple, banana, cherry} is the same as {cherry, banana, apple}.

2. Formula: The number of ways to choose $k$ items from $n$ items (not considering the order) can be calculated using the formula for combinations, often referred to as "n choose k":

   $$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$

3. Factorial:

   • The factorial of a number $n$, denoted $n!$, is the product of all positive integers up to $n$.
   • For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.

Examples

1. Simple Example:

   • If you want to choose 2 fruits out of 4: an apple, a banana, a cherry, and a date, you can calculate the number of combinations using the formula:

     $$\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3 \times 2 \times 1}{(2 \times 1)(2 \times 1)} = \frac{24}{4} = 6$$

2. Real-World Example:

   • Consider a lottery where you need to pick 6 numbers out of 49. The number of possible combinations is:

     $$\binom{49}{6} = \frac{49!}{6!(49-6)!}$$

     Expanding and simplifying the factorials will give you the total number of unique combinations.

Applications

• Statistics: Used to calculate probabilities when the order of events does not matter.
• Computer Science: Utilized in algorithms that deal with grouping or partitioning data.
• Everyday Decisions: Helps in making choices like forming teams, creating playlists, or selecting menu items where the sequence is unimportant.

Understanding combinations allow us to solve a wide range of problems involving selection and arrangement without worrying about permutations where the order is significant.

Concept of Ratio of Favorable Outcomes to Total Outcomes

In probability, the ratio of favorable outcomes to total outcomes is a fundamental concept used to determine the likelihood of a specific event occurring.

The favorable outcomes are the outcomes that satisfy the event in question, while the total outcomes are all possible outcomes in the given sample space.

Definition and Formula

The probability of an event, denoted by $P(E)$, is calculated using the ratio of the number of favorable outcomes $n(E)$ to the number of total outcomes in the sample space $n(S)$. This ratio is given by the formula:

Example Calculation

Suppose we roll a fair six-sided die. The sample space $S$ consists of the outcomes: $\{1, 2, 3, 4, 5, 6\}$. If we are interested in the event $E$ of rolling an even number, the favorable outcomes are $\{2, 4, 6\}$.

• Number of favorable outcomes $n(E) = 3$
• Total number of outcomes $n(S) = 6$

Using the formula for probability:

Importance in Probability

Understanding this ratio is crucial because it forms the basis for more advanced probability concepts and calculations. It helps quantify how likely an event is to happen in a clear and straightforward manner.

In summary, the ratio of favorable outcomes to total outcomes is not only an essential concept in probability theory but also serves as the cornerstone for predicting and analyzing random events in various fields such as statistics, gaming, risk assessment, and more.