Are $$ A $$ and $$ B $$ independent events? Two events $$ A $$ and $$ B $$ are not independent because the occurrence of one affects the chance the other occurs. Numerically, $$ P(A \mid B)
eq P(A) $$, so events $$ A $$ and $$ B $$ are not independent. Two events $$ A $$ and $$ B $$ are independent because the occurrence of one does not affect the chance the other occurs. Numerically, $$ P(A \mid B)=P(A) $$, so events $$ A $$ and $$ B $$ are independent.

Question

Are $$ A $$ and $$ B $$ independent events? Two events $$ A $$ and $$ B $$ are not independent because the occurrence of one affects the chance the other occurs. Numerically, $$ P(A \mid B) 
eq P(A) $$, so events $$ A $$ and $$ B $$ are not independent. Two events $$ A $$ and $$ B $$ are independent because the occurrence of one does not affect the chance the other occurs. Numerically, $$ P(A \mid B)=P(A) $$, so events $$ A $$ and $$ B $$ are independent.

UpStudy AI · Accepted Answer

To determine whether two events $$ A $$ and $$ B $$ are independent, we examine how the occurrence of one event affects the probability of the other. Let's break down the concepts you've provided:

### **Definitions of Independence and Dependence**

1. **Dependent Events:**
   - **Statement:** *"Two events $$ A $$ and $$ B $$ are not independent because the occurrence of one affects the chance the other occurs. Numerically, $$ P(A \mid B) \neq P(A) $$, so events $$ A $$ and $$ B $$ are not independent."*
   - **Explanation:** This means that knowing $$ B $$ has occurred changes the probability of $$ A $$ occurring. For example, if $$ B $$ is "it is raining" and $$ A $$ is "the ground is wet," then $$ P(A \mid B) $$ (probability the ground is wet given that it's raining) is likely higher than $$ P(A) $$ (probability the ground is wet without that knowledge).

2. **Independent Events:**
   - **Statement:** *"Two events $$ A $$ and $$ B $$ are independent because the occurrence of one does not affect the chance the other occurs. Numerically, $$ P(A \mid B) = P(A) $$, so events $$ A $$ and $$ B $$ are independent."*
   - **Explanation:** Here, knowing that $$ B $$ occurred doesn't change the probability of $$ A $$ occurring. For instance, if $$ B $$ is "rolling a 2 on a six-sided die" and $$ A $$ is "flipping heads on a coin," then $$ P(A \mid B) = P(A) $$ because the outcome of the die doesn't influence the coin flip.

### **Formal Definition of Independence**

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

$$
P(A \mid B) = P(A)
$$

**Alternatively**, this can be expressed as:

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

Both formulations mean that the occurrence of one event has no effect on the probability of the other.

### **Determining Independence**

To assess whether $$ A $$ and $$ B $$ are independent:

1. **Calculate $$ P(A) $$, $$ P(B) $$, and $$ P(A \cap B) $$:**
   - If $$ P(A \cap B) = P(A) \times P(B) $$, then $$ A $$ and $$ B $$ are independent.
   - If not, they are dependent.

2. **Using Conditional Probability:**
   - If $$ P(A \mid B) = P(A) $$, then $$ A $$ and $$ B $$ are independent.
   - If $$ P(A \mid B) \neq P(A) $$, then they are dependent.

### **Examples**

1. **Independent Events Example:**
   - **Event A:** Rolling a 4 on a six-sided die.
   - **Event B:** Getting tails on a coin flip.
   - **Calculation:** $$ P(A) = \frac{1}{6} $$, $$ P(B) = \frac{1}{2} $$, $$ P(A \cap B) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} $$.
   - Since $$ P(A \cap B) = P(A) \times P(B) $$, events $$ A $$ and $$ B $$ are independent.

2. **Dependent Events Example:**
   - **Event A:** Drawing an Ace from a standard deck of cards.
   - **Event B:** Drawing a Spade from the same deck.
   - **Calculation:** 
     - $$ P(A) = \frac{4}{52} $$
     - $$ P(B) = \frac{13}{52} $$
     - $$ P(A \cap B) = \frac{1}{52} $$ (only the Ace of Spades satisfies both)
     - $$ P(A) \times P(B) = \frac{4}{52} \times \frac{13}{52} = \frac{52}{2704} = \frac{1}{52} $$
   - Interestingly, in this case, $$ P(A \cap B) = P(A) \times P(B) $$, suggesting independence. However, since knowing that a card is a Spade slightly alters the probability of it being an Ace (from $$ \frac{4}{52} $$ to $$ \frac{1}{13} $$), these events are actually **not** independent in practical scenarios. This highlights the importance of context and accurate probability calculations.

### **Conclusion**

To determine if events $$ A $$ and $$ B $$ are independent:

- **Independent:** $$ P(A \mid B) = P(A) $$ or $$ P(A \cap B) = P(A) \times P(B) $$
- **Dependent:** $$ P(A \mid B) \neq P(A) $$ or $$ P(A \cap B) \neq P(A) \times P(B) $$

Use these criteria to analyze the relationship between any two events based on their probabilities.
Events $$ A $$ and $$ B $$ are **dependent** because the occurrence of one affects the probability of the other.

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