# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Mutually Exclusive Events A and B",
  description: "Master 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.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/21/2025",
  subject: "Probability",
};

## 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.

## 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.

### Probability of Event A AND B Occurring Together

Because events A and B 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:

<BlockMath math="P(A \text{ and } B) = 0" />

Or using the intersection symbol:

<BlockMath math="P(A \cap B) = 0" />

Remember, if they are mutually exclusive, their intersection is empty, so the probability is zero!

## Calculating the Probability of A OR B 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?

Since they can't happen together, we simply **add** the probabilities of each individual event.

The formula becomes very easy:

<BlockMath math="P(A \text{ or } B) = P(A) + P(B)" />

Or using the union symbol:

<BlockMath 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?

    - <InlineMath math="P(\text{Heads}) = 1/2" />
    - <InlineMath math="P(\text{Tails}) = 1/2" />
    - Probability of getting Heads OR Tails = <InlineMath math="P(\text{Heads}) + P(\text{Tails}) = 1/2 + 1/2 = 1" /> (One of them must occur).

2.  **Rolling a Die (once):**

    - The event of getting a "3" and the event of getting a "5". You can't get both numbers at once.
      - <InlineMath math="P(3) = 1/6" />
      - <InlineMath math="P(5) = 1/6" />
      - Probability of getting a 3 OR 5 = <InlineMath math="P(3) + P(5) = 1/6 + 1/6 = 2/6 = 1/3" />.
    - The event of getting an "even number" (<InlineMath math="\{2, 4, 6\}" />) and the event of getting an "odd number" (<InlineMath math="\{1, 3, 5\}" />). No number is both even and odd.
      - <InlineMath math="P(\text{Even}) = 3/6 = 1/2" />
      - <InlineMath math="P(\text{Odd}) = 3/6 = 1/2" />
      - Probability of getting Even OR Odd = <InlineMath math="P(\text{Even}) + P(\text{Odd}) = 1/2 + 1/2 = 1" />.

3.  **Drawing a Card (once):**

    - The event of getting a "King" and the event of getting a "Queen". One card cannot be both a King and a Queen.
      - There are 4 Kings in 52 cards, <InlineMath math="P(\text{King}) = 4/52" />
      - There are 4 Queens in 52 cards, <InlineMath math="P(\text{Queen}) = 4/52" />
      - Probability of getting a King OR Queen = <InlineMath math="P(\text{King}) + P(\text{Queen}) = 4/52 + 4/52 = 8/52 = 2/13" />.
    - The event of getting a "Red card (Hearts/Diamonds)" and the event of getting a "Club (<InlineMath math="\clubsuit" />)". Clubs are black, so a card cannot be red and a club simultaneously.
      - There are 26 red cards, <InlineMath math="P(\text{Red}) = 26/52" />
      - There are 13 clubs, <InlineMath math="P(\clubsuit) = 13/52" />
      - Probability of getting Red OR Club = <InlineMath math="P(\text{Red}) + P(\clubsuit) = 26/52 + 13/52 = 39/52 = 3/4" />.
