# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/probability/probability-distribution
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/probability/probability-distribution/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Probability Distribution",
  description: "Understand probability distributions with dice examples. Learn sample spaces, event calculations, and real-world applications through interactive scenarios.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/21/2025",
  subject: "Probability",
};

## What is a Probability Distribution Anyway?

Imagine you're playing a game of rolling dice. A probability distribution is like a complete list of all possible outcomes when you roll the dice, plus it tells you how big the chance (probability) is for each outcome to appear. Simply put, it's a list of possibilities and how often they happen.

## Getting to Know the Sample Space (All Possible Outcomes)

The sample space is like the collection of all outcomes that _can_ happen in an experiment. We usually write it using the symbol <InlineMath math="\Omega" /> (Omega).

For example, if you roll a standard die with <InlineMath math="6" /> sides:
The numbers that can show up are <InlineMath math="1, 2, 3, 4, 5," /> or <InlineMath math="6" />. Well, all these possibilities gathered together are called the sample space.

<BlockMath math="\Omega = \{1, 2, 3, 4, 5, 6\}" />

There are a total of <InlineMath math="6" /> possible outcomes here.

## Events and Their Probabilities

### What is an Event?

An event is one or more _specific_ outcomes that we are interested in from the sample space. An event is a smaller part of the sample space.

**Example**:

From rolling one die earlier (<InlineMath math="\Omega = \{1, 2, 3, 4, 5, 6\}" />), let's say we want to look at the event "getting an even number".
The even numbers there are <InlineMath math="2, 4," /> and <InlineMath math="6" />. So, the event of getting an even number (let's call it event A) is:

<BlockMath math="A = \{2, 4, 6\}" />

There are <InlineMath math="3" /> outcomes in this event A.

### Calculating the Probability of an Event

Probability is a number that shows how likely an event is to happen. Calculating it is easy:

<BlockMath math="P(\text{Event}) = \frac{\text{Number of outcomes in that event}}{\text{Total number of outcomes in the sample space}}" />

Using the symbols from our example:

<BlockMath math="P(A) = \frac{|A|}{|\Omega|}" />

Where:

- <InlineMath math="|A|" /> means the number of outcomes in event A (there were <InlineMath math="3" />
  , right?).
- <InlineMath math="|\Omega|" /> means the total number of outcomes in the sample
  space (there were <InlineMath math="6" />, right?).

So, the probability of the event "getting an even number" (event A) is:

<BlockMath math="P(A) = \frac{3}{6} = \frac{1}{2}" />

This means the chance is half-and-half, or <InlineMath math="50\%" />.

## Rules of the Game for Probability Distributions

Probability distributions have two important rules:

1.  The probability of each outcome (<InlineMath math="P(x)" />) must be a value between <InlineMath math="0" /> and <InlineMath math="1" />. It can't be negative or greater than <InlineMath math="1" />.

    <BlockMath math="0 \leq P(x) \leq 1" />

2.  If you add up all the probabilities for every outcome in the sample space, the total must be exactly <InlineMath math="1" />.

    <BlockMath math="\sum_{x \in \Omega} P(x) = 1" />

## Rolling One Die

If we roll one fair die (meaning each side has an equal chance of showing up), the distribution looks like this:

| Outcome <InlineMath math="x" /> | Probability <InlineMath math="P(x)" /> |
| :------------------------------ | :------------------------------------- |
| <InlineMath math="1" />         | <InlineMath math="1/6" />              |
| <InlineMath math="2" />         | <InlineMath math="1/6" />              |
| <InlineMath math="3" />         | <InlineMath math="1/6" />              |
| <InlineMath math="4" />         | <InlineMath math="1/6" />              |
| <InlineMath math="5" />         | <InlineMath math="1/6" />              |
| <InlineMath math="6" />         | <InlineMath math="1/6" />              |

**Why are they all <InlineMath math="1/6" />?**

Because there are <InlineMath math="6" /> sides, and the die is fair, so each side has <InlineMath math="1" /> chance out of the total <InlineMath math="6" /> possibilities. Try adding up all the probabilities: <InlineMath math="1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 = 1" />. It fits the second rule perfectly!

## Rolling Two Dice

Now imagine rolling two dice, say **one red die and one white die**. If we list all the possible pairs of numbers that can appear, there will be <InlineMath math="36" /> possibilities!

**Why <InlineMath math="36" />?**

Because the first die has <InlineMath math="6" /> possibilities, the second die also has <InlineMath math="6" />, so the total is <InlineMath math="6 \times 6 = 36" /> pairs.

The resulting pairs can be written like this:

- Red Die 1, White Die 1 -> (1, 1)
- Red Die 1, White Die 2 -> (1, 2)
- ... and so on until ...
- Red Die 6, White Die 6 -> (6, 6)

Each of these pairs has an equally small probability, which is <InlineMath math="1/36" />.

**Important! Distinguish between (1, 2) and (2, 1)!**

If the dice are different colors (like red and white), getting a <InlineMath math="3" /> on the red die and a <InlineMath math="2" /> on the white die is different from getting a <InlineMath math="2" /> on the red die and a <InlineMath math="3" /> on the white die. So, the order matters if the dice can be distinguished.

### Probability Distribution for the Sum of Two Dice

Often, we are interested in the _sum_ of the numbers on the two dice. The smallest sum is <InlineMath math="1+1=2" />, and the largest is <InlineMath math="6+6=12" />. The probability for each sum varies:

| Sum of Numbers <InlineMath math="j" /> | Possible Pairs                                                 | Number of Pairs         | Probability <InlineMath math="P(j)" /> |
| :------------------------------------- | :------------------------------------------------------------- | :---------------------- | :------------------------------------- |
| <InlineMath math="2" />                | <InlineMath math="(1,1)" />                                    | <InlineMath math="1" /> | <InlineMath math="1/36" />             |
| <InlineMath math="3" />                | <InlineMath math="(1,2), (2,1)" />                             | <InlineMath math="2" /> | <InlineMath math="2/36" />             |
| <InlineMath math="4" />                | <InlineMath math="(1,3), (2,2), (3,1)" />                      | <InlineMath math="3" /> | <InlineMath math="3/36" />             |
| <InlineMath math="5" />                | <InlineMath math="(1,4), (2,3), (3,2), (4,1)" />               | <InlineMath math="4" /> | <InlineMath math="4/36" />             |
| <InlineMath math="6" />                | <InlineMath math="(1,5), (2,4), (3,3), (4,2), (5,1)" />        | <InlineMath math="5" /> | <InlineMath math="5/36" />             |
| <InlineMath math="7" />                | <InlineMath math="(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)" /> | <InlineMath math="6" /> | <InlineMath math="6/36" />             |
| <InlineMath math="8" />                | <InlineMath math="(2,6), (3,5), (4,4), (5,3), (6,2)" />        | <InlineMath math="5" /> | <InlineMath math="5/36" />             |
| <InlineMath math="9" />                | <InlineMath math="(3,6), (4,5), (5,4), (6,3)" />               | <InlineMath math="4" /> | <InlineMath math="4/36" />             |
| <InlineMath math="10" />               | <InlineMath math="(4,6), (5,5), (6,4)" />                      | <InlineMath math="3" /> | <InlineMath math="3/36" />             |
| <InlineMath math="11" />               | <InlineMath math="(5,6), (6,5)" />                             | <InlineMath math="2" /> | <InlineMath math="2/36" />             |
| <InlineMath math="12" />               | <InlineMath math="(6,6)" />                                    | <InlineMath math="1" /> | <InlineMath math="1/36" />             |

Look, the sum <InlineMath math="7" /> is the most common outcome (<InlineMath math="6/36" />), while sums <InlineMath math="2" /> and <InlineMath math="12" /> are the rarest (<InlineMath math="1/36" />). This is very important in many board games!

## Why is Learning Probability Distribution Important?

Probability distributions are useful for many things:

- **Knowing what's more likely:** We can see which outcomes have the biggest chance of happening and which ones have the smallest. Like knowing that a sum of 7 is more likely than a sum of 12.
- **Playing games:** Many games (like Monopoly, Snakes and Ladders, etc.) use dice. Understanding probability distributions can help us make better strategies (although there's still luck involved!).
- **Simple predictions:** It can help us make educated guesses. For example, knowing the weather distribution can help predict the chance of rain tomorrow.
- **Making decisions:** In business or science, probability distributions are used to make decisions based on data, so we don't just guess randomly.

In short, probability distributions help us understand uncertainty using numbers!
