Let S be a set of outcomes and let A and B be events with outcomes in S. Let simB...

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## Understanding the Question

Let's clarify what we're looking for. We need to find **the exact value of P(A)** - the probability that event A occurs.

**Given Information:**
- S is our universe of all possible outcomes
- A and B are events within S
- ~B represents everything in S that's NOT in B (the complement of B)
- We need a specific numerical value for P(A)

**What "Sufficient" Means:** A statement (or combination) is sufficient if it allows us to determine one unique value for P(A). If multiple values are possible, it's NOT sufficient.

**Key Insight:** This is a "coverage" problem. Since B and ~B together make up all of S, the unions A∪B and A∪~B will overlap only at A. This insight will be crucial when we combine statements.

## Analyzing Statement 1

**What Statement 1 Tells Us:** P(A ∪ B) = 0.7

This means 70% of all outcomes are in either A or B (or both). But this single equation has too many unknowns—we don't know P(A), P(B), or how much they overlap.

Let's test specific scenarios to see if different values of P(A) are possible:

**Scenario 1:** What if B is empty (P(B) = 0)?
- Then P(A ∪ B) = P(A) = 0.7
- So P(A) = 0.7 ✓

**Scenario 2:** What if A and B don't overlap at all and are equal in size?
- If P(A) = P(B) = 0.35 and P(A ∩ B) = 0
- Then P(A ∪ B) = 0.35 + 0.35 = 0.7 ✓
- So P(A) = 0.35 ✓

Since we found two different possible values for P(A) (0.7 and 0.35), Statement 1 is **NOT sufficient**.

**[STOP - Not Sufficient!]** This eliminates choices A and D.

## Analyzing Statement 2

**Now let's forget Statement 1 completely and analyze Statement 2 independently.**

**What Statement 2 Provides:** P(A ∪ ~B) = 0.9

This tells us that 90% of outcomes are either in A or NOT in B (or both). Again, we have one equation with multiple unknowns.

Let's test scenarios:

**Scenario 1:** What if B contains half of S (P(B) = 0.5)?
- Then P(~B) = 0.5
- If A is entirely within B with no overlap with ~B:
  - P(A ∪ ~B) = P(A) + P(~B) = P(A) + 0.5 = 0.9
  - This gives P(A) = 0.4 ✓

**Scenario 2:** What if B is very small (P(B) = 0.1)?
- Then P(~B) = 0.9
- If A is entirely within B (no overlap with ~B):
  - P(A ∪ ~B) = P(~B) = 0.9
  - This means P(A) could be any value from 0 to 0.1 ✓

Different values of P(A) are possible, so Statement 2 is **NOT sufficient**.

**[STOP - Not Sufficient!]** This eliminates choice B.

## Combining Statements

**Combined Information:**
- Statement 1: P(A ∪ B) = 0.7
- Statement 2: P(A ∪ ~B) = 0.9

Here's where our key insight becomes powerful. Let's think about what these two unions actually cover:

1. **A ∪ B covers:** All of B plus any part of A outside B
2. **A ∪ ~B covers:** All of ~B plus any part of A inside B

Since B and ~B together make up all of S (they partition S), these two unions together must cover everything in S. But here's the crucial point: **A appears in both unions!**

When we add the probabilities:
- We count everything in B exactly once (in the first union)
- We count everything in ~B exactly once (in the second union)  
- We count everything in A **twice** (once in each union)

Therefore: 
P(A ∪ B) + P(A ∪ ~B) = P(S) + P(A) = 1 + P(A)

Substituting our values:
0.7 + 0.9 = 1 + P(A)
1.6 = 1 + P(A)
**P(A) = 0.6**

We get exactly one unique value! The statements together are **sufficient**.

**[STOP - Sufficient!]** This eliminates choice E.

## The Answer: C

Both statements together give us exactly one value for P(A), but neither statement alone is sufficient.

**Answer Choice C**: "Both statements together are sufficient, but neither statement alone is sufficient."