# 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/combinatorics/probability-of-an-event Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/combinatorics/probability-of-an-event/en.mdx Learn probability of an event with clear formulas, sample space concepts, and real-world examples. Learn to calculate event likelihoods. --- ## 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: Visible text: **Probability** is a measure of the likelihood of an event occurring. Probability gives a numerical value between and , 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 Visible text: - Probability means the event is **impossible** to occur - Probability means the event will **certainly** occur - Probability 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 $$\{AA, AG, GA, GG\}$$, where $$A$$ represents heads and $$G$$ represents tails. If we want to know the probability of getting at least one tail, then event $$A = \{AG, GA, GG\}$$. Visible text: **Concrete example:** when flipping two coins simultaneously, the sample space is , where represents heads and represents tails. If we want to know the probability of getting at least one tail, then event . ## Probability Formula To calculate the probability of an event, we use the **classical probability formula**: ```math P(A) = \frac{n(A)}{n(S)} ``` Where: - $$P(A)$$ = probability of event $$A$$ occurring - $$n(A)$$ = number of favorable outcomes (members of event $$A$$) - $$n(S)$$ = number of all possible outcomes (members of sample space) Visible text: - = probability of event occurring - = number of favorable outcomes (members of event ) - = 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:** - $$0 \leq P(A) \leq 1$$ (probability is always between $$0$$ and $$1$$) - , where is the **complement** of event $$A$$ Visible text: - (probability is always between and ) - , where is the **complement** of event **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 is "getting an odd number". Visible text: **Complement Concept:** The complement of event is all outcomes in the sample space that are **not** included in event . For example, if is "getting an even number", then is "getting an odd number". ## Applications in Real Situations **Analysis of Rolling Two Dice:** When rolling two dice, the total possible outcomes are $$6 \times 6 = 36$$ pairs. Let's analyze the probability of getting a sum of $$9$$: Visible text: When rolling two dice, the total possible outcomes are pairs. Let's analyze the probability of getting a sum of : **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: $$n(A) = 4$$ 3. Determine probability: ```math P(\text{sum} = 9) = \frac{4}{36} = \frac{1}{9} \approx 0.111 ``` Visible text: 1. Identify all ways to get sum : - First die , second die : - First die , second die : - First die , second die : - First die , second die : 2. Count the number of favorable events: 3. Determine probability: **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 $$\frac{3}{32}$$. Visible text: 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 . **Practical interpretation:** Out of every $$32$$ milk boxes produced, an average of $$3$$ boxes contain prizes. This information helps the company: Visible text: **Practical interpretation:** Out of every milk boxes produced, an average of 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: Visible text: In traditional dice games, players often use probability understanding to make decisions. For example, the probability of getting sum (which occurs most frequently) is: Ways to get sum $$7$$: $$(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 \text{ ways}$$ Visible text: Ways to get sum : ```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 $$5 \text{ red balls}$$, $$3 \text{ blue balls}$$, and $$2 \text{ green balls}$$. 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 $$1000 \text{ units}$$ 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? Visible text: 1. In rolling two dice, determine the probability of getting a sum that is an even number. 2. A box contains , , and . 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 of product. Based on experience, 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: $$\frac{1}{2}$$** **Systematic solution steps:** Sample space for rolling two dice: $$n(S) = 36$$ **Identify all even sums and ways to obtain them:** - Sum $$2$$: $$(1,1)$$ → $$1$$ way - Sum $$4$$: $$(1,3), (2,2), (3,1)$$ → $$3 \text{ ways}$$ - Sum $$6$$: $$(1,5), (2,4), (3,3), (4,2), (5,1)$$ → $$5 \text{ ways}$$ - Sum $$8$$: $$(2,6), (3,5), (4,4), (5,3), (6,2)$$ → $$5 \text{ ways}$$ - Sum $$10$$: $$(4,6), (5,5), (6,4)$$ → $$3 \text{ ways}$$ - Sum $$12$$: $$(6,6)$$ → $$1$$ way **Total favorable events:** $$n(A) = 1 + 3 + 5 + 5 + 3 + 1 = 18$$ **Probability calculation:** ```math P(\text{even}) = \frac{18}{36} = \frac{1}{2} = 0.5 ``` 2. **Answer: $$\frac{1}{2}$$** **Solution steps:** **Count total balls:** $$5 + 3 + 2 = 10$$ balls **Identify balls that are not red:** $$3 \text{ blue} + 2 \text{ green} = 5$$ balls **Probability calculation:** ```math P(\text{not red}) = \frac{5}{10} = \frac{1}{2} = 0.5 ``` 3. **Answer: $$\frac{3}{8}$$** **Solution steps:** **Sample space for three coin flips:** $$\{AAA, AAG, AGA, AGG, GAA, GAG, GGA, GGG\}$$ **Total possibilities:** $$n(S) = 2^3 = 8$$ **Event exactly two tails:** $$\{AGG, GAG, GGA\}$$ **Number of favorable events:** $$n(A) = 3$$ **Probability calculation:** ```math P(\text{exactly 2 tails}) = \frac{3}{8} = 0.375 ``` 4. **Answer: $$\frac{19}{20}$$ or $$95\%$$** **Solution steps:** **Total products:** $$1000 \text{ units}$$ **Defective products:** $$5\% \times 1000 = 50 \text{ units}$$ **Non-defective products:** $$1000 - 50 = 950 \text{ units}$$ **Probability calculation:** ```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. Visible text: 1. **Answer: ** **Systematic solution steps:** Sample space for rolling two dice: **Identify all even sums and ways to obtain them:** - Sum : → way - Sum : → - Sum : → - Sum : → - Sum : → - Sum : → way **Total favorable events:** **Probability calculation:** 2. **Answer: ** **Solution steps:** **Count total balls:** balls **Identify balls that are not red:** balls **Probability calculation:** 3. **Answer: ** **Solution steps:** **Sample space for three coin flips:** **Total possibilities:** **Event exactly two tails:** **Number of favorable events:** **Probability calculation:** 4. **Answer: or ** **Solution steps:** **Total products:** **Defective products:** **Non-defective products:** **Probability calculation:** **Interpretation:** There is a probability that the product purchased by a consumer is not defective.