# Nakafa Learning Content

> For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`.

URL: https://nakafa.com/en/subjects/mathematics/data-analysis-probability/uniform-distribution
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/data-analysis-probability/uniform-distribution/en.mdx

Learn uniform distribution with equal probability examples using dice and cards. Understand f(x)=1/k formula and solve probability problems one step at a time.

---

## Understanding Simple Probability

In everyday life, we often face situations with uncertain outcomes. A simple example is when flipping a coin. There are two possible outcomes: heads or tails. Well, if the coin is balanced and there's no cheating, then both sides have the same chance of appearing. This concept is the foundation of probability.

Mathematically, the probability of an event $$A$$ can be calculated in a fairly simple way: just divide the number of desired outcomes by the total number of all possible outcomes.

Visible text: Mathematically, the probability of an event can be calculated in a fairly simple way: just divide the number of desired outcomes by the total number of all possible outcomes.

```math
P(A) = \frac{n(A)}{n(S)}
```

## Concept of Uniform Distribution

Uniform distribution is actually the simplest concept in the world of probability. Its characteristic is that **every possible outcome has exactly the same probability** of occurring. Imagine dividing a pizza into equal slices for everyone - no one gets a bigger or smaller piece.

Let's take an example of a fair six-sided die. The numbers that can appear are $$1, 2, 3, 4, 5, 6$$. The probability of getting a $$1$$ is exactly the same as the probability of getting a $$6$$, which is one out of six possibilities. No number is "luckier" than the others. This is what we mean by uniform distribution - all outcomes have the same chance.

Visible text: Let's take an example of a fair six-sided die. The numbers that can appear are . The probability of getting a is exactly the same as the probability of getting a , which is one out of six possibilities. No number is "luckier" than the others. This is what we mean by uniform distribution - all outcomes have the same chance.

## Mathematical Formula

To express uniform distribution mathematically, we have a very simple formula. If a random variable $$X$$ can produce $$k$$ different possibilities, and each possibility has the same probability, then the formula is:

Visible text: To express uniform distribution mathematically, we have a very simple formula. If a random variable can produce different possibilities, and each possibility has the same probability, then the formula is:

```math
f(x;k) = \frac{1}{k}
```

Where:

- $$f(x;k)$$ is the probability function for a specific outcome when there are $$k$$ total possibilities

- $$x$$ is one of the possible outcomes (for example, the number $$3$$ on a die)

- $$k$$ is the total number of possible outcomes (for example, $$6$$ for a die)

Visible text: - is the probability function for a specific outcome when there are total possibilities

- is one of the possible outcomes (for example, the number on a die)

- is the total number of possible outcomes (for example, for a die)

> In other words, the probability for each outcome is **one divided by the total number of possible outcomes**. Pretty easy, right?

## Practical Applications

Now let's see how this concept works in real situations.

### Die Rolling

For example, we have a balanced six-sided die that is rolled once. How do we determine its uniform distribution?

- **Possible Outcomes:** The sample space is $$\{1, 2, 3, 4, 5, 6\}$$

- **Total Number of Outcomes:** There are $$6$$ possible outcomes, so $$k=6$$

- **Distribution Function:** Using our formula:

  
  
  ```math
  f(x;6) = \frac{1}{6}
  ```

- **Meaning:** The probability of getting the numbers $$1, 2, 3, 4, 5, 6$$ is each $$\frac{1}{6}$$ or approximately $$16{,}67\%$$

Visible text: - **Possible Outcomes:** The sample space is 

- **Total Number of Outcomes:** There are possible outcomes, so 

- **Distribution Function:** Using our formula:

 
 

- **Meaning:** The probability of getting the numbers is each or approximately

### Prize Wheel

Another example, suppose there's a spinning wheel divided into $$8$$ equal sections, each numbered $$1$$ to $$8$$. What's the probability that the wheel stops on number $$5$$?

Visible text: Another example, suppose there's a spinning wheel divided into equal sections, each numbered to . What's the probability that the wheel stops on number ?

- **Possible Outcomes:** $$\{1, 2, 3, 4, 5, 6, 7, 8\}$$

- **Total Number of Outcomes:** There are $$8$$ sections, so $$k=8$$

- **Distribution Function:**

  
  
  ```math
  f(x;8) = \frac{1}{8}
  ```

- **Meaning:** The probability that the wheel stops on number $$5$$ (or any other number) is $$\frac{1}{8}$$ or $$12{,}5\%$$

Visible text: - **Possible Outcomes:** 

- **Total Number of Outcomes:** There are sections, so 

- **Distribution Function:**

 
 

- **Meaning:** The probability that the wheel stops on number (or any other number) is or

## Exercises

1. A bag contains $$10$$ identical balls except for their colors: $$1 \text{ red ball}$$, $$1 \text{ blue ball}$$, $$1 \text{ green ball}$$, and so on up to $$10$$ different colors. If one ball is drawn randomly, what is the probability of drawing a yellow ball? Also write the uniform distribution function.

2. In a standard bridge card set containing $$52 \text{ cards}$$, each card has the same probability of being drawn. What can you conclude about the probability distribution of drawing one card from that deck? What is the probability of drawing the King of Hearts?

Visible text: 1. A bag contains identical balls except for their colors: , , , and so on up to different colors. If one ball is drawn randomly, what is the probability of drawing a yellow ball? Also write the uniform distribution function.

2. In a standard bridge card set containing , each card has the same probability of being drawn. What can you conclude about the probability distribution of drawing one card from that deck? What is the probability of drawing the King of Hearts?

### Answer Key

1. **Answer for Colored Ball Problem**

    - **Step** $$1$$: Calculate total possibilities.

      There are $$10 \text{ balls}$$ with different colors, so the total possible outcomes is $$k=10$$

    - **Step** $$2$$: Determine the uniform distribution function.

      Since each ball has the same probability of being drawn (no ball is "special"), we use the uniform distribution formula:

      
      
      ```math
      f(x;10) = \frac{1}{10}
      ```

      Where $$x$$ represents one of the $$10 \text{ ball}$$ colors available

    - **Step** $$3$$: Calculate the probability of the yellow ball.

      The yellow ball is one of the $$10 \text{ balls}$$, so its probability is the same as other colored balls:

      
      
      ```math
      P(\text{Yellow Ball}) = \frac{1}{10}
      ```

      Therefore, the probability of drawing a yellow ball is **$$\frac{1}{10}$$** or **$$10\%$$**

2. **Answer for Bridge Card Problem**

    - **Step** $$1$$: Analyze the type of distribution.

      Since each card has the same probability of being drawn (no card is easier to draw), the probability distribution is **uniform distribution**

    - **Step** $$2$$: Calculate total possibilities.

      A standard bridge card set has $$52$$ different cards, so $$k=52$$

    - **Step** $$3$$: Calculate the probability of King of Hearts.

      The King of Hearts is one of the $$52 \text{ cards}$$ available, so using the uniform distribution principle:

      
      
      ```math
      P(\text{King of Hearts}) = \frac{1}{52}
      ```

      Therefore, the probability of drawing the King of Hearts is **$$\frac{1}{52}$$** or approximately **$$1{,}92\%$$**

Visible text: 1. **Answer for Colored Ball Problem**

 - **Step** : Calculate total possibilities.

 There are with different colors, so the total possible outcomes is 

 - **Step** : Determine the uniform distribution function.

 Since each ball has the same probability of being drawn (no ball is "special"), we use the uniform distribution formula:

 
 

 Where represents one of the colors available

 - **Step** : Calculate the probability of the yellow ball.

 The yellow ball is one of the , so its probability is the same as other colored balls:

 
 

 Therefore, the probability of drawing a yellow ball is **** or ****

2. **Answer for Bridge Card Problem**

 - **Step** : Analyze the type of distribution.

 Since each card has the same probability of being drawn (no card is easier to draw), the probability distribution is **uniform distribution**

 - **Step** : Calculate total possibilities.

 A standard bridge card set has different cards, so 

 - **Step** : Calculate the probability of King of Hearts.

 The King of Hearts is one of the available, so using the uniform distribution principle:

 
 

 Therefore, the probability of drawing the King of Hearts is **** or approximately ****