To solve the problem, we need to analyze both statements regarding the events A, B, and C.
### Step 1: Understand the Definitions
First, we need to understand what mutually exclusive events are. Two or more events are said to be mutually exclusive if the occurrence of one event means that none of the other events can occur at the same time. Mathematically, this means:
- \( P(A \cap B) = 0 \)
- \( P(A \cap C) = 0 \)
- \( P(B \cap C) = 0 \)
### Step 2: Evaluate Statement 1
We are given:
- \( P(A) = \frac{2}{3} \)
- \( P(B) = \frac{1}{4} \)
- \( P(C) = \frac{1}{6} \)
To check if A, B, and C are mutually exclusive, we need to see if the sum of their probabilities is less than or equal to 1:
\[
P(A) + P(B) + P(C) = \frac{2}{3} + \frac{1}{4} + \frac{1}{6}
\]
To add these fractions, we find a common denominator. The least common multiple of 3, 4, and 6 is 12.
Converting each fraction:
- \( P(A) = \frac{2}{3} = \frac{8}{12} \)
- \( P(B) = \frac{1}{4} = \frac{3}{12} \)
- \( P(C) = \frac{1}{6} = \frac{2}{12} \)
Now, adding them together:
\[
P(A) + P(B) + P(C) = \frac{8}{12} + \frac{3}{12} + \frac{2}{12} = \frac{13}{12}
\]
Since \( \frac{13}{12} > 1 \), it indicates that A, B, and C cannot be mutually exclusive because the total probability exceeds 1.
### Conclusion for Statement 1
Thus, **Statement 1 is false**.
### Step 3: Evaluate Statement 2
Statement 2 states that if \( P(A \cup B \cup C) = P(A) + P(B) + P(C) \), then A, B, and C are mutually exclusive.
Using the formula for the probability of the union of three events:
\[
P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)
\]
If A, B, and C are mutually exclusive, then:
\[
P(A \cap B) = P(B \cap C) = P(C \cap A) = P(A \cap B \cap C) = 0
\]
Thus, the equation simplifies to:
\[
P(A \cup B \cup C) = P(A) + P(B) + P(C)
\]
Since we have already calculated \( P(A) + P(B) + P(C) = \frac{13}{12} \), if \( P(A \cup B \cup C) \) also equals \( \frac{13}{12} \), then it confirms that A, B, and C are mutually exclusive.
### Conclusion for Statement 2
Since the condition holds true, **Statement 2 is true**.
### Final Answer
- Statement 1: False
- Statement 2: True
