What does it mean for two events to be independent in probability?

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In probability theory, **independence** between two events means that the occurrence or non-occurrence of one event does not influence the probability of the other event occurring. In other words, knowing that one event has occurred provides no information about whether the other event will occur.

### Formal Definition

Two events, $$ A $$ and $$ B $$, are **independent** if and only if:

$$
P(A \cap B) = P(A) \times P(B)
$$

- $$ P(A \cap B) $$ is the probability that both events $$ A $$ and $$ B $$ occur.
- $$ P(A) $$ and $$ P(B) $$ are the probabilities of events $$ A $$ and $$ B $$ occurring individually.

### Alternative Characterizations

Independence can also be expressed in terms of conditional probability:

- **Conditional Probability Approach:**
  
  $$
  P(A \mid B) = P(A) \quad \text{and} \quad P(B \mid A) = P(B)
  $$
  
  This means the probability of $$ A $$ occurring given that $$ B $$ has occurred is the same as the probability of $$ A $$ occurring without any knowledge about $$ B $$, and vice versa.

### Example

**Scenario:** Consider flipping a fair six-sided die and flipping a fair coin.

- Let event $$ A $$ be "rolling a 4" on the die.
- Let event $$ B $$ be "getting heads" on the coin.

**Calculations:**

- $$ P(A) = \frac{1}{6} $$ (since there's one favorable outcome out of six).
- $$ P(B) = \frac{1}{2} $$ (since there's one favorable outcome out of two).
- $$ P(A \cap B) = P(\text{rolling a 4 and getting heads}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} $$.

Since $$ P(A \cap B) = P(A) \times P(B) $$, events $$ A $$ and $$ B $$ are independent. The outcome of the die roll does not affect the outcome of the coin flip, and vice versa.

### Why Independence Matters

Understanding whether events are independent is crucial in probability because it simplifies the calculation of combined probabilities. If events are independent, you can easily find the probability of both occurring together by multiplying their individual probabilities. However, if events are **dependent**, the occurrence of one event affects the probability of the other, requiring different methods to compute their joint probability.

### Common Misconceptions

- **Mutual Exclusivity vs. Independence:**
  
  It's important not to confuse independence with mutual exclusivity. **Mutually exclusive** events cannot occur at the same time (e.g., flipping a coin resulting in both heads and tails simultaneously), and for mutually exclusive events (with non-zero probabilities), they are **not independent** because the occurrence of one guarantees the non-occurrence of the other.

- **Multiple Events:**
  
  Independence can extend to more than two events, but it's defined pairwise and collectively, meaning every subset of events must satisfy the independence condition.

### Summary

Two events are independent if the occurrence of one does not affect the probability of the other. This concept is foundational in probability theory and is essential for accurately modeling and analyzing random processes where events do not influence each other.
Two events are independent in probability if the occurrence of one does not affect the probability of the other occurring.

What does it mean for two events to be independent in probability?

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