# 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/two-events-mutually-exclusive Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/probability/two-events-mutually-exclusive/en.mdx Learn mutually exclusive events that cannot occur together. Learn P(A or B) = P(A) + P(B) formula with coin, dice, and card examples plus calculations. --- ## What Does Mutually Exclusive Mean? Imagine you have two choices, but you can only pick **one**, not both at the same time. Well, in the world of probability, this is similar to the concept of **Mutually Exclusive Events** (other cool names: _Disjoint Events_). Two events (let's call them event $$A$$ and event $$B$$) are said to be **mutually exclusive** if both events **cannot possibly occur at the same time** in a single trial. Simply put, if $$A$$ happens, $$B$$ cannot happen. If $$B$$ happens, $$A$$ cannot happen. Visible text: Two events (let's call them event and event ) are said to be **mutually exclusive** if both events **cannot possibly occur at the same time** in a single trial. Simply put, if happens, cannot happen. If happens, cannot happen. ## Characteristics of Mutually Exclusive Events The main characteristic is just that: **cannot happen simultaneously**. There's no outcome that can belong to event $$A$$ and event $$B$$ at the same time. Visible text: The main characteristic is just that: **cannot happen simultaneously**. There's no outcome that can belong to event and event at the same time. ### Probability of Two Events Occurring Together Because events $$A$$ and $$B$$ cannot happen together if they are mutually exclusive, the probability of both occurring simultaneously is **zero**! Visible text: Because events and cannot happen together if they are mutually exclusive, the probability of both occurring simultaneously is **zero**! We can write the probability of event "$$A$$ **and** $$B$$" (both occurring) as: Visible text: We can write the probability of event " **and** " (both occurring) as: ```math P(A \text{ and} B) = 0 ``` Or using the intersection symbol: ```math P(A \cap B) = 0 ``` Remember, if they are mutually exclusive, their intersection is empty, so the probability is zero! ## Calculating the Combined Probability for Mutually Exclusive Events So, how do we calculate the probability of event $$A$$ happening **OR** event $$B$$ happening if $$A$$ and $$B$$ are mutually exclusive? Visible text: So, how do we calculate the probability of event happening **OR** event happening if and are mutually exclusive? Since they can't happen together, we simply **add** the probabilities of each individual event. The formula becomes very easy: ```math P(A \text{ or} B) = P(A) + P(B) ``` Or using the union symbol: ```math P(A \cup B) = P(A) + P(B) ``` This is the **Special Addition Rule** which applies ONLY to mutually exclusive events. (If the events are not mutually exclusive, there's a slightly different formula). ## Examples of Mutually Exclusive Events To understand better, look at these examples: 1. **Coin Toss:** The event of getting "Heads" and the event of getting "Tails". They can't happen together, right? - $$P(\text{Heads}) = 1/2$$ - $$P(\text{Tails}) = 1/2$$ - Probability of getting Heads OR Tails is $$P(\text{Heads}) + P(\text{Tails}) = 1/2 + 1/2 = 1$$ (One of them must occur). 2. **Rolling a Die (once):** Look at two pairs of events: - Getting a $$3$$ and getting a $$5$$ cannot happen at once. Each has probability $$1/6$$, so the probability of getting a $$3$$ or $$5$$ is $$1/6 + 1/6 = 2/6 = 1/3$$. - Getting an even number $$\{2, 4, 6\}$$ and getting an odd number $$\{1, 3, 5\}$$ cannot happen at once. Each event has probability $$3/6 = 1/2$$, so the probability of getting even or odd is $$1/2 + 1/2 = 1$$. 3. **Drawing a Card (once):** Use the same idea for card events: - Getting a King and getting a Queen are mutually exclusive. There are $$4$$ Kings and $$4$$ Queens in $$52 \text{ cards}$$, so the probability of getting a King or Queen is $$4/52 + 4/52 = 8/52 = 2/13$$. - Getting a red card and getting a club $$\clubsuit$$ are mutually exclusive because clubs are black. There are $$26$$ red cards and $$13$$ clubs, so the probability of getting red or club is $$26/52 + 13/52 = 39/52 = 3/4$$. Visible text: 1. **Coin Toss:** The event of getting "Heads" and the event of getting "Tails". They can't happen together, right? - - - Probability of getting Heads OR Tails is (One of them must occur). 2. **Rolling a Die (once):** Look at two pairs of events: - Getting a and getting a cannot happen at once. Each has probability , so the probability of getting a or is . - Getting an even number and getting an odd number cannot happen at once. Each event has probability , so the probability of getting even or odd is . 3. **Drawing a Card (once):** Use the same idea for card events: - Getting a King and getting a Queen are mutually exclusive. There are Kings and Queens in , so the probability of getting a King or Queen is . - Getting a red card and getting a club are mutually exclusive because clubs are black. There are red cards and clubs, so the probability of getting red or club is .