Determine whether the following individual events are overlapping or​ non-overlapping. Then find the probability of the combined event.
Getting a sum of either 5​, 7​, or 9 on a roll of two dice
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Part 1
Choose the correct answer below​ and, if​ necessary, fill in the answer box to complete your choice.
​(Type an integer or a simplified​ fraction.)
A.
The individual events are​ non-overlapping. The probability of the combined event is 
  
enter your response here.
B.
The individual events are overlapping. The probability of the combined event is 
  
enter your response here.

To solve this, let's break the problem down step-by-step:

### 1. Identify the possible sums on two dice:
When rolling two dice, the possible sums range from 2 to 12. We are particularly interested in the sums of 5, 7, and 9.

### 2. List the possible ways to get the sums of 5, 7, and 9:
- **Sum of 5**: (1, 4), (2, 3), (3, 2), (4, 1) → 4 outcomes
- **Sum of 7**: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1) → 6 outcomes
- **Sum of 9**: (3, 6), (4, 5), (5, 4), (6, 3) → 4 outcomes

### 3. Check if the events overlap:
There are no repeated outcomes between the events that give sums of 5, 7, and 9. Therefore, the events are **non-overlapping**.

### 4. Calculate the total number of outcomes for the combined event:
We simply add the number of outcomes for each event:
- 4 outcomes for a sum of 5
- 6 outcomes for a sum of 7
- 4 outcomes for a sum of 9

Total outcomes for the combined event = \( 4 + 6 + 4 = 14 \)

### 5. Calculate the probability:
There are a total of 36 possible outcomes when rolling two dice (since each die has 6 faces, and \(6 \times 6 = 36\)).

The probability of getting a sum of either 5, 7, or 9 is the ratio of favorable outcomes to the total outcomes:
\[
P(\text{sum of 5, 7, or 9}) = \frac{14}{36} = \frac{7}{18}
\]

### Conclusion:
- The events are **non-overlapping**.
- The probability of the combined event is \( \frac{7}{18} \).

### Final Answer:
**A. The individual events are non-overlapping. The probability of the combined event is \( \frac{7}{18} \).**

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Do you want any details or have questions about this?  
Here are 5 related questions:
1. How do you calculate the probability of mutually exclusive events?
2. What are dependent and independent events in probability?
3. How do you find the total possible outcomes when rolling more than two dice?
4. How does the probability change if a dice is weighted?
5. What is the probability of rolling a sum of 2 on two dice?

**Tip:** When dealing with dice rolls, listing all possible outcomes can help in determining probabilities quickly.

Determine whether the following individual events are overlapping or non-overlapping. Then find the probability of the combined event. Getting a sum of either 5, 7, or 9 on a roll of two dice.

Discover how to determine whether events are overlapping or non-overlapping and how to calculate the probability of getting a sum of 5, 7, or 9 when rolling two dice. Learn the step-by-step breakdown of outcomes and probability, suitable for students in Grades 6-8.

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