# 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/probability/probability-distribution Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/probability/probability-distribution/en.mdx Understand probability distributions with dice examples. Learn sample spaces, event calculations, and real-world applications through interactive scenarios. --- ## 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 $$\Omega$$ (Omega). Visible text: The sample space is like the collection of all outcomes that _can_ happen in an experiment. We usually write it using the symbol (Omega). For example, if you roll a standard die with $$6$$ sides: The numbers that can show up are $$1, 2, 3, 4, 5,$$ or $$6$$. Well, all these possibilities gathered together are called the sample space. Visible text: For example, if you roll a standard die with sides: The numbers that can show up are or . Well, all these possibilities gathered together are called the sample space. ```math \Omega = \{1, 2, 3, 4, 5, 6\} ``` There are a total of $$6$$ possible outcomes here. Visible text: There are a total of 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 ($$\Omega = \{1, 2, 3, 4, 5, 6\}$$), let's say we want to look at the event "getting an even number". The even numbers there are $$2, 4,$$ and $$6$$. So, the event of getting an even number (let's call it event $$A$$) is: Visible text: From rolling one die earlier (), let's say we want to look at the event "getting an even number". The even numbers there are and . So, the event of getting an even number (let's call it event ) is: ```math A = \{2, 4, 6\} ``` There are $$3$$ outcomes in this event $$A$$. Visible text: There are outcomes in this event . ### Calculating the Probability of an Event Probability is a number that shows how likely an event is to happen. Calculating it is easy: ```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: ```math P(A) = \frac{|A|}{|\Omega|} ``` Where: - $$|A|$$ means the number of outcomes in event $$A$$ (there were $$3$$ , right?). - $$|\Omega|$$ means the total number of outcomes in the sample space (there were $$6$$, right?). Visible text: - means the number of outcomes in event (there were , right?). - means the total number of outcomes in the sample space (there were , right?). So, the probability of the event "getting an even number" (event $$A$$) is: Visible text: So, the probability of the event "getting an even number" (event ) is: ```math P(A) = \frac{3}{6} = \frac{1}{2} ``` This means the chance is half-and-half, or $$50\%$$. Visible text: This means the chance is half-and-half, or . ## Rules of the Game for Probability Distributions Probability distributions have two important rules: 1. The probability of each outcome ($$P(x)$$) must be a value between $$0$$ and $$1$$. It can't be negative or greater than $$1$$. ```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 $$1$$. ```math \sum_{x \in \Omega} P(x) = 1 ``` Visible text: 1. The probability of each outcome () must be a value between and . It can't be negative or greater than . 2. If you add up all the probabilities for every outcome in the sample space, the total must be exactly . ## 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 $$x$$ | Probability $$P(x)$$ | | :------------------------------ | :------------------------------------- | | $$1$$ | $$1/6$$ | | $$2$$ | $$1/6$$ | | $$3$$ | $$1/6$$ | | $$4$$ | $$1/6$$ | | $$5$$ | $$1/6$$ | | $$6$$ | $$1/6$$ | Visible text: | Outcome | Probability | | :------------------------------ | :------------------------------------- | | | | | | | | | | | | | | | | | | | **Why are they all $$1/6$$?** Visible text: **Why are they all ?** Because there are $$6$$ sides, and the die is fair, so each side has $$1$$ chance out of the total $$6$$ possibilities. Try adding up all the probabilities: $$1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 = 1$$. It fits the second rule perfectly! Visible text: Because there are sides, and the die is fair, so each side has chance out of the total possibilities. Try adding up all the probabilities: . 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 $$36$$ possibilities! Visible text: 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 possibilities! **Why $$36$$?** Visible text: **Why ?** Because the first die has $$6$$ possibilities, the second die also has $$6$$, so the total is $$6 \times 6 = 36$$ pairs. Visible text: Because the first die has possibilities, the second die also has , so the total is pairs. The resulting pairs can be written like this: - Red die $$1$$, white die $$1$$ gives $$(1, 1)$$. - Red die $$1$$, white die $$2$$ gives $$(1, 2)$$. - ... and so on until ... - Red die $$6$$, white die $$6$$ gives $$(6, 6)$$. Visible text: - Red die , white die gives . - Red die , white die gives . - ... and so on until ... - Red die , white die gives . Each of these pairs has an equally small probability, which is $$1/36$$. Visible text: Each of these pairs has an equally small probability, which is . **Important! Distinguish between $$(1, 2)$$ and $$(2, 1)$$!** Visible text: **Important! Distinguish between and !** If the dice are different colors (like red and white), getting a $$3$$ on the red die and a $$2$$ on the white die is different from getting a $$2$$ on the red die and a $$3$$ on the white die. So, the order matters if the dice can be distinguished. Visible text: If the dice are different colors (like red and white), getting a on the red die and a on the white die is different from getting a on the red die and a 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 $$1+1=2$$, and the largest is $$6+6=12$$. The probability for each sum varies: Visible text: Often, we are interested in the _sum_ of the numbers on the two dice. The smallest sum is , and the largest is . The probability for each sum varies: | Sum of Numbers $$j$$ | Possible Pairs | Number of Pairs | Probability $$P(j)$$ | | :------------------------------------- | :------------------------------------------------------------- | :---------------------- | :------------------------------------- | | $$2$$ | $$(1,1)$$ | $$1$$ | $$1/36$$ | | $$3$$ | $$(1,2), (2,1)$$ | $$2$$ | $$2/36$$ | | $$4$$ | $$(1,3), (2,2), (3,1)$$ | $$3$$ | $$3/36$$ | | $$5$$ | $$(1,4), (2,3), (3,2), (4,1)$$ | $$4$$ | $$4/36$$ | | $$6$$ | $$(1,5), (2,4), (3,3), (4,2), (5,1)$$ | $$5$$ | $$5/36$$ | | $$7$$ | $$(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$$ | $$6$$ | $$6/36$$ | | $$8$$ | $$(2,6), (3,5), (4,4), (5,3), (6,2)$$ | $$5$$ | $$5/36$$ | | $$9$$ | $$(3,6), (4,5), (5,4), (6,3)$$ | $$4$$ | $$4/36$$ | | $$10$$ | $$(4,6), (5,5), (6,4)$$ | $$3$$ | $$3/36$$ | | $$11$$ | $$(5,6), (6,5)$$ | $$2$$ | $$2/36$$ | | $$12$$ | $$(6,6)$$ | $$1$$ | $$1/36$$ | Visible text: | Sum of Numbers | Possible Pairs | Number of Pairs | Probability | | :------------------------------------- | :------------------------------------------------------------- | :---------------------- | :------------------------------------- | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Look, the sum $$7$$ is the most common outcome ($$6/36$$), while sums $$2$$ and $$12$$ are the rarest ($$1/36$$). This is very important in many board games! Visible text: Look, the sum is the most common outcome (), while sums and are the rarest (). 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. Visible text: - **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 is more likely than a sum of . - **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!