E. Chapter 3 HW- Probability Topics e. Are $$ A $$ and $$ B $$ mutually exclusive events? It is possible to have a four or a five on the first die rolling followed by an odd number on the second die when the sum of these two numbers is at A most eight. Numerically, we have $$ P(A $$ AND $$ B)=\frac{1}{9} $$, so the events are not mutually exclusive. It is impossible to have a four or a five on the first die rolling followed by an odd number on the second die when the sum of these two numbers is at most eight. Numerically, we have $$ P(A $$ AND $$ B)=0 $$, so the events are mutually exclusive. f. 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, B $$ P(A \mid B)=P(A) $$, so events $$ A $$ and $$ B $$ are independent.

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E. Chapter 3 HW- Probability Topics e. Are $$ A $$ and $$ B $$ mutually exclusive events? It is possible to have a four or a five on the first die rolling followed by an odd number on the second die when the sum of these two numbers is at A most eight. Numerically, we have $$ P(A $$ AND $$ B)=\frac{1}{9} $$, so the events are not mutually exclusive. It is impossible to have a four or a five on the first die rolling followed by an odd number on the second die when the sum of these two numbers is at most eight. Numerically, we have $$ P(A $$ AND $$ B)=0 $$, so the events are mutually exclusive. f. 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, B $$ P(A \mid B)=P(A) $$, so events $$ A $$ and $$ B $$ are independent.

UpStudy AI · Accepted Answer

To determine whether events $$ A $$ and $$ B $$ are mutually exclusive and independent, we need to analyze the definitions and the provided probabilities.

### Part e: Are $$ A $$ and $$ B $$ mutually exclusive events?

**Definition of Mutually Exclusive Events:**
Two events $$ A $$ and $$ B $$ are mutually exclusive if they cannot occur at the same time. This means that the probability of both events occurring together, $$ P(A \text{ AND } B) $$, must be equal to 0.

**Given Information:**
1. $$ P(A \text{ AND } B) = \frac{1}{9} $$ (indicating that both events can occur together).
2. Another scenario states $$ P(A \text{ AND } B) = 0 $$ (indicating that both events cannot occur together).

**Conclusion:**
- If $$ P(A \text{ AND } B) = \frac{1}{9} $$, then $$ A $$ and $$ B $$ are **not mutually exclusive**.
- If $$ P(A \text{ AND } B) = 0 $$, then $$ A $$ and $$ B $$ are **mutually exclusive**.

Thus, the answer depends on the specific scenario being considered. If the first scenario is true, then they are not mutually exclusive. If the second scenario is true, then they are mutually exclusive.

### Part f: Are $$ A $$ and $$ B $$ independent events?

**Definition of Independent Events:**
Two events $$ A $$ and $$ B $$ are independent if the occurrence of one does not affect the occurrence of the other. This can be mathematically expressed as:
$$
P(A \mid B) = P(A)
$$
or equivalently,
$$
P(B \mid A) = P(B)
$$

**Given Information:**
1. If $$ P(A \mid B) \neq P(A) $$, then $$ A $$ and $$ B $$ are **not independent**.
2. If $$ P(A \mid B) = P(A) $$, then $$ A $$ and $$ B $$ are **independent**.

**Conclusion:**
- If $$ P(A \mid B) \neq P(A) $$, then $$ A $$ and $$ B $$ are not independent.
- If $$ P(A \mid B) = P(A) $$, then $$ A $$ and $$ B $$ are independent.

### Summary:
- For part e, the mutual exclusivity of events $$ A $$ and $$ B $$ depends on the specific probability given.
- For part f, the independence of events $$ A $$ and $$ B $$ also depends on the relationship between $$ P(A \mid B) $$ and $$ P(A) $$.

If you have specific values for $$ P(A) $$, $$ P(B) $$, and $$ P(A \mid B) $$, we can further analyze the situation.
Events $$ A $$ and $$ B $$ are **not mutually exclusive** because $$ P(A \text{ AND } B) = \frac{1}{9} $$. They are **not independent** because $$ P(A \mid B) \neq P(A) $$.

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