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Mutually exclusive events in probability theory are events that cannot occur at the same time, such as flipping a coin to get heads or tails. This concept is crucial for calculating probabilities and is represented using Venn diagrams and set theory. The addition rule for these events helps solve real-life probability problems by summing individual probabilities. Differentiating them from independent events is vital for accurate probability analysis.

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## Definition of Mutually Exclusive Events

### Key concept in probability theory

Mutually exclusive events are events that cannot occur at the same time, essential for accurately calculating probabilities

### Examples of Mutually Exclusive Events

Flipping a fair coin

When flipping a fair coin, the result can only be heads or tails, not both, making these outcomes mutually exclusive

Drawing a single card from a deck

In drawing a single card from a deck, it cannot be both an ace and a queen, making these outcomes mutually exclusive

### Representations of Mutually Exclusive Events

Venn diagrams

Venn diagrams depict mutually exclusive events as non-intersecting circles, with no shared area

Set theory

In set theory, mutually exclusive events are denoted as A ∩ B = ∅, indicating no common outcomes

## Addition Rule for Mutually Exclusive Events

### Fundamental concept for calculating probabilities

The addition rule states that the probability of the occurrence of any one of the mutually exclusive events is the sum of their individual probabilities

### Real-world applications of the Addition Rule

Rolling a die

The addition rule can be applied to calculate the probability of rolling a 1 or a 2 on a fair six-sided die

Drawing cards from a deck

The addition rule can be used to calculate the probability of drawing a heart or a club from a standard deck of cards

## Distinction between Mutually Exclusive and Independent Events

### Different properties and implications for probability

Mutually exclusive events cannot occur at the same time, while independent events have no influence on each other's probability

### Calculation of probabilities for each type of event

Independent events

The joint probability of independent events is the product of their individual probabilities

Mutually exclusive events

The combined probability of mutually exclusive events is the sum of their individual probabilities

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