# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/combinatorics/probability-of-an-event
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/combinatorics/probability-of-an-event/en.mdx

Output docs content for large language models.

---

export const metadata = {
   title: "Probability of an Event",
   description: "Master probability of an event with clear formulas, sample space concepts, and real-world examples. Learn to calculate event likelihoods effectively.",
   authors: [{ name: "Nabil Akbarazzima Fatih" }],
   date: "05/26/2025",
   subject: "Combinatorics",
};

## Understanding Probability

Have you ever wondered about the chances of rain today? Or what's the probability of your favorite team winning a match? In daily life, we often face uncertain situations, and this is where the concept of **probability** becomes important.

**Probability** is a measure of the likelihood of an event occurring. Probability gives a numerical value between 0 and 1, where:
- Probability 0 means the event is **impossible** to occur
- Probability 1 means the event will **certainly** occur
- Probability 0.5 means the event has an **equal chance** of occurring or not

The concept of probability helps us make decisions based on data and analysis, not just based on intuition alone.

## Sample Space and Events

Before calculating probability, we need to understand two fundamental concepts:

**Sample Space (S)** is the set of all possible outcomes from an experiment. For example, when rolling two dice, the sample space consists of all possible pairs of numbers that can appear.

**Event (A)** is a subset of the sample space that represents the outcome we want or the outcome we are examining.

**Concrete example:** when flipping two coins simultaneously, the sample space is <InlineMath math="\{AA, AG, GA, GG\}" />, where A = Heads and G = Tails. If we want to know the probability of getting at least one tail, then event A = <InlineMath math="\{AG, GA, GG\}" />.

## Probability Formula

To calculate the probability of an event, we use the **classical probability formula**:

<BlockMath math="P(A) = \frac{n(A)}{n(S)}" />

Where:
- <InlineMath math="P(A)" /> = probability of event A occurring
- <InlineMath math="n(A)" /> = number of favorable outcomes (members of event A)
- <InlineMath math="n(S)" /> = number of all possible outcomes (members of sample space)

**Important condition:** This formula applies when all outcomes in the sample space have **equal probability** of occurring (equiprobable).

**Fundamental properties of probability:**
- <InlineMath math="0 \leq P(A) \leq 1" /> (probability is always between 0 and 1)
- <InlineMath math="P(A) + P(A') = 1" />, where <InlineMath math="A'" /> is the **complement** of event A

**Complement Concept:** The complement of event A is all outcomes in the sample space that are **not** included in event A. For example, if A is "getting an even number", then A' is "getting an odd number".

## Applications in Real Situations

**Analysis of Rolling Two Dice:**

When rolling two dice, the total possible outcomes are <InlineMath math="6 \times 6 = 36" /> pairs. Let's analyze the probability of getting a sum of 9:

**Systematic steps:**

1. Identify all ways to get sum 9:
   - First die 3, second die 6: (3,6)
   - First die 4, second die 5: (4,5)  
   - First die 5, second die 4: (5,4)
   - First die 6, second die 3: (6,3)

2. Count the number of favorable events: <InlineMath math="n(A) = 4" />

3. Determine probability: 
    <BlockMath math="P(\text{sum} = 9) = \frac{4}{36} = \frac{1}{9} \approx 0.111" />

**Probability-Based Marketing Strategy:**

A beverage company runs a prize program by inserting coupons in each milk box. Based on historical data, the probability of someone buying a milk box containing a prize is <InlineMath math="\frac{3}{32}" />.

**Practical interpretation:** Out of every 32 milk boxes produced, an average of 3 boxes contain prizes. This information helps the company:
- Plan promotional budgets
- Estimate consumer response  
- Determine sales targets

**Strategic Gaming:**

In traditional dice games, players often use probability understanding to make decisions. For example, the probability of getting sum 7 (which occurs most frequently) is:

Ways to get sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 ways

<BlockMath math="P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6} \approx 0.167" />

This is the highest probability compared to other sums, so it's often used as the basis for game strategy.

## Practice Problems

1. In rolling two dice, determine the probability of getting a sum that is an even number.

2. A box contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn randomly, determine the probability of getting a ball that is not red.

3. In flipping three coins simultaneously, determine the probability of getting exactly two tails.

4. A company produces 1000 units of product. Based on experience, 5% of these products are defective. If a consumer buys one product randomly, what is the probability that the product is not defective?

### Answer Key

1. **Answer: <InlineMath math="\frac{1}{2}" />**
   
   **Systematic solution steps:**
   
   Sample space for rolling two dice: <InlineMath math="n(S) = 36" />
   
   **Identify all even sums and ways to obtain them:**
   - Sum 2: (1,1) → 1 way
   - Sum 4: (1,3), (2,2), (3,1) → 3 ways  
   - Sum 6: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 ways
   - Sum 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 ways
   - Sum 10: (4,6), (5,5), (6,4) → 3 ways
   - Sum 12: (6,6) → 1 way
   
   **Total favorable events:** <InlineMath math="n(A) = 1 + 3 + 5 + 5 + 3 + 1 = 18" />
   
   **Probability calculation:**
   <BlockMath math="P(\text{even}) = \frac{18}{36} = \frac{1}{2} = 0.5" />

2. **Answer: <InlineMath math="\frac{1}{2}" />**
   
   **Solution steps:**
   
   **Count total balls:** <InlineMath math="5 + 3 + 2 = 10" /> balls
   
   **Identify balls that are not red:** <InlineMath math="3 \text{ blue} + 2 \text{ green} = 5" /> balls
   
   **Probability calculation:**
   <BlockMath math="P(\text{not red}) = \frac{5}{10} = \frac{1}{2} = 0.5" />

3. **Answer: <InlineMath math="\frac{3}{8}" />**
   
   **Solution steps:**
   
   **Sample space for three coin flips:** <InlineMath math="\{AAA, AAG, AGA, AGG, GAA, GAG, GGA, GGG\}" />
   
   **Total possibilities:** <InlineMath math="n(S) = 2^3 = 8" />
   
   **Event exactly two tails:** <InlineMath math="\{AGG, GAG, GGA\}" />
   
   **Number of favorable events:** <InlineMath math="n(A) = 3" />
   
   **Probability calculation:**
   <BlockMath math="P(\text{exactly 2 tails}) = \frac{3}{8} = 0.375" />

4. **Answer: <InlineMath math="\frac{19}{20}" /> or 95%**
   
   **Solution steps:**
   
   **Total products:** 1000 units
   
   **Defective products:** <InlineMath math="5\% \times 1000 = 50" /> units
   
   **Non-defective products:** <InlineMath math="1000 - 50 = 950" /> units
   
   **Probability calculation:**
   <BlockMath math="P(\text{not defective}) = \frac{950}{1000} = \frac{19}{20} = 0.95 = 95\%" />
   
   **Interpretation:** There is a 95% probability that the product purchased by a consumer is not defective.