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Let's go through each part step-by-step.

### Given Information:
- There are **7 tiles**, labeled with the first 7 letters of the alphabet: **A, B, C, D, E, F, G**.
- Each outcome has an **equal probability**.

### Definitions of Events:
1. **Event X**: Selecting a letter that comes **before E**. 
   - Letters before E are **A, B, C, and D**.
2. **Event Y**: Selecting a **consonant** (non-vowel).
   - Consonants among A-G are **B, C, D, F, and G**.

### (a) Identifying Outcomes and Probabilities:
1. **Event X (letters before E)**:
   - Outcomes: **A, B, C, D**
   - Probability \( P(X) = \frac{4}{7} \)

2. **Event Y (consonants)**:
   - Outcomes: **B, C, D, F, G**
   - Probability \( P(Y) = \frac{5}{7} \)

3. **Event X or Y (letters before E or consonants)**:
   - Outcomes for X: **A, B, C, D**
   - Outcomes for Y: **B, C, D, F, G**
   - Combined unique outcomes for \( X \text{ or } Y \): **A, B, C, D, F, G** (6 outcomes)
   - Probability \( P(X \text{ or } Y) = \frac{6}{7} \)

4. **Event X and Y (letters before E and consonants)**:
   - Intersection outcomes of X and Y: **B, C, D**
   - Probability \( P(X \text{ and } Y) = \frac{3}{7} \)

### (b) Compute \( P(X) + P(Y) - P(X \text{ and } Y) \):
   - Substitute values:
     \[
     P(X) + P(Y) - P(X \text{ and } Y) = \frac{4}{7} + \frac{5}{7} - \frac{3}{7} = \frac{6}{7}
     \]

### (c) Select the Answer:
The computed value \( \frac{6}{7} \) makes the equation true.

Would you like a detailed breakdown of any specific part, or do you have further questions?

### Follow-Up Questions:
1. How do you calculate the probability of combined events (like \( X \text{ or } Y \))?
2. Why is the probability of the union of events calculated using inclusion-exclusion?
3. How does identifying overlapping outcomes help in probability calculations?
4. What is the probability of choosing a vowel in this set?
5. If each letter had a different probability, how would calculations change?

### Tip:
When dealing with probabilities of "or" events, remember to use the inclusion-exclusion principle to avoid double-counting overlapping outcomes.

A tile is selected at random from 7 tiles labeled with the first 7 letters of the alphabet. Complete parts (a) through (c) by determining probabilities of certain events and computing probability equations.

This problem explores probability concepts by calculating the likelihood of selecting certain tiles labeled A-G. Understand event outcomes, use the inclusion-exclusion principle, and solve for probabilities like P(X or Y) and P(X and Y). Ideal for students in Grades 9-11.

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