A discrete probability distribution requires two essential criteria. Firstly, all probabilities of outcomes must be non-negative, indicating that the probability of an event occurring is never negative. Secondly, the sum of all probabilities over the entire sample space must equal 1. This ensures that at least one of the possible outcomes will occur when the experiment is performed. The non-negativity and normalization requirements guarantee the validity and reliability of a discrete probability distribution.

##
- Define discrete probability distributions and their importance in probability theory.

Probability theory is an essential branch of mathematics that deals with the analysis of random phenomena. **Discrete probability distributions** play a crucial role in this field by providing a mathematical framework for modeling the probabilities of specific outcomes in discrete random experiments.

Imagine tossing a fair coin: there are only two possible outcomes, heads or tails. Each outcome has an equal chance of occurring. **Discrete probability distributions** allow us to quantify this chance of occurrence for each outcome.

**Significance of Discrete Probability Distributions**

In real-life scenarios, many phenomena can be represented as discrete random experiments, such as:

- Counting the number of customers arriving at a store in a given hour
- Determining the number of defective items in a batch of products
- Predicting the outcome of a roulette spin

**Discrete probability distributions** provide accurate models to analyze these experiments, allowing us to make informed decisions and predictions about the future. They are used in a wide range of applications, including statistics, finance, and engineering.

Understanding discrete probability distributions forms a foundational step in the exploration of probability theory. By mastering these concepts, you will gain essential insights into the behavior of random phenomena and learn to predict outcomes with confidence.

## The Sample Space: A Foundation for Probability

In the realm of probability, the concept of a **sample space** is paramount. It’s the cornerstone upon which all probability calculations rest, setting the boundaries of possible outcomes and forming the basis for determining the likelihood of specific events.

At its core, the sample space is a set that encompasses **all possible outcomes** of a given experiment or random event. It’s the *universal set*, the umbrella under which every possible result exists.

**Elements** of the sample space are the individual outcomes, the fundamental units that make up the universe of possibilities. Consider the classic example of rolling a six-sided die: the sample space consists of six elements, each representing the number that can appear on the top face (1, 2, 3, 4, 5, 6).

The **cardinality** of the sample space, denoted by n, refers to the **number of elements** it contains. In the die-rolling scenario, the cardinality is six. Understanding the cardinality is crucial because it determines the total possible number of outcomes.

To summarize, the sample space provides a **structured framework** for analyzing probability. It defines the complete set of potential outcomes, allowing us to assign probabilities to specific events and make informed predictions about the uncertain nature of random phenomena.

## Outcomes: Elementary Events and Sample Points – The Indivisible Building Blocks of Probability

In the realm of probability theory, the **sample space** represents the universe of all possible outcomes for an experiment or event. Within this sample space, we encounter two fundamental concepts: **elementary events** and **sample points**. These indivisible building blocks form the foundation for understanding the intricacies of probability distributions.

Imagine tossing a fair coin. The sample space for this experiment consists of two outcomes: heads or tails. Each of these outcomes is an **elementary event**, an indivisible outcome that cannot be further subdivided. We can represent the sample space as a set: {H, T}, where H denotes heads and T denotes tails.

**Sample points**, on the other hand, refer to specific instances within the sample space. For our coin toss, we have two sample points: one for heads and one for tails. These sample points are denoted as h and t, respectively. The set of sample points is equivalent to the sample space: {h, t}.

Understanding elementary events and sample points is crucial for comprehending how probabilities are assigned. In the case of our coin toss, the probability of getting heads is equal to the number of favorable sample points (1) divided by the total number of sample points (2). This gives us a probability of 0.5, indicating that heads has an equal chance of occurring as tails.

By understanding the concepts of elementary events and sample points, we lay the groundwork for exploring the intricacies of probability distributions. These fundamental building blocks provide a clear framework for modeling and analyzing the outcomes of various experiments and events, helping us better understand the world around us.

## Events: Subsets and Relationships in Discrete Probability Distributions

As we delve deeper into the intricate world of discrete probability distributions, we encounter the concept of **events**, which are simply subsets of the **sample space**, the union of all possible outcomes. Events play a crucial role in understanding the probability of specific outcomes and the relationships between them.

Consider a sample space consisting of possible outcomes when flipping a coin: {Heads, Tails}. An event could be “**obtaining Heads**“. This event is represented as a subset of the sample space: {Heads}. Another event could be “**not obtaining Heads**“, which is equivalent to {Tails}.

**Complementary events** are two events that together span the entire sample space. In our coin toss example, “obtaining Heads” and “not obtaining Heads” are complementary events, as they cover all possible outcomes.

**Mutually exclusive events** are events that cannot occur simultaneously. Suppose we roll two dice and observe the sum of the numbers on the top faces. The event “the sum is 7” is mutually exclusive with the event “the sum is 11”, because the same pair of numbers cannot produce both sums simultaneously.

Understanding these relationships is essential for comprehending the behavior of discrete probability distributions and applying them to real-world scenarios. These concepts underpin the analysis of phenomena involving uncertainty and randomness, enabling us to make informed decisions and predict future outcomes.

**Probability Mass Function**

- Introduce the probability mass function as a function assigning probabilities to outcomes.
- Explain the difference between probability density function and cumulative distribution function.

**Probability Mass Function: Assigning Probabilities to Discrete Outcomes**

In the realm of probability, discrete probability distributions serve as indispensable tools for analyzing the likelihood of specific outcomes in scenarios where each possible outcome has a **distinct and countable value**. At the heart of these distributions lies the probability mass function, a central concept that assigns probabilities to each of the possible outcomes within the sample space.

Imagine rolling a six-sided die. The sample space for this experiment comprises the numbers 1 through 6. Each number represents an **elementary event**, an indivisible outcome of the experiment. Assigning probabilities to these outcomes allows us to predict the likelihood of obtaining a particular number when rolling the die.

The **probability mass function (PMF)** is a function that **explicitly assigns a probability to each outcome** in the sample space. It is typically denoted as (P(X = x) ), where (X) represents the random variable representing the outcome and (x) represents a specific outcome within the sample space.

For instance, in the case of a six-sided die, the PMF might assign a probability of 1/6 to each of the six outcomes (1, 2, 3, 4, 5, 6). This means that each outcome has an equal chance of occurring when the die is rolled.

**Distinguishing PMF from Other Probability Functions**

It is crucial to differentiate the probability mass function from two other commonly used probability functions: the **probability density function** and the **cumulative distribution function**. The probability density function is used for continuous random variables, where outcomes can take on any value within a specified range. The cumulative distribution function, on the other hand, provides the probability of obtaining an outcome less than or equal to a specified value.

In contrast, the probability mass function is solely applicable to discrete random variables, where outcomes are restricted to a set of distinct values. It assigns probabilities to these individual outcomes, making it an essential tool for understanding the likelihood of specific events occurring in discrete probability experiments.

**Requirements for a Discrete Probability Distribution**

- State the non-negativity requirement, ensuring probabilities are non-negative.
- Explain normalization, ensuring the sum of probabilities equals 1.
- Discuss additivity, representing the probabilities of disjoint events.

**Non-Negativity Requirement**

Imagine a world where probabilities could be negative. It would be a chaotic realm where the likelihood of an event happening could be expressed as a negative number. Fortunately, we don’t live in that bizarre universe. The *non-negativity requirement* states that all probabilities must be *non-negative.* This means that the chance of an event occurring can never be less than zero.

**Normalization: The Sum of All Probabilities Equals 1**

Picture a huge lottery with countless possible outcomes. Each ticket represents a specific outcome, and the sum of all winning tickets should equal the *total probability of winning.* This is where *normalization* comes in. It ensures that the sum of all probabilities in a discrete probability distribution is 1. It’s like a giant puzzle where every piece must fit together perfectly, accounting for all possible outcomes.

**Additivity: Combining Disjointed Events**

Imagine two independent events, like flipping a coin and rolling a die. If you want to find the probability of both events occurring, you can use *additivity.* This principle states that the probability of the union of *disjoint events* (events that cannot happen simultaneously) is equal to the sum of their individual probabilities. So, the probability of flipping a coin and rolling a 6 on a die is simply the sum of the probabilities of flipping heads and rolling a 6.