Let S = {1, 2, 3, 4, 5, 6} and let P be such that P({n}) = 1/6 for each n ∈ S. Give an
example of two independent events A, B ⊆ S such that P(A) ̸ = 0, P(A) ̸ = 1, P(B) ̸ = 0,
P(B) ̸ = 1

We are given a sample space \( S = \{1, 2, 3, 4, 5, 6\} \) and a probability measure \( P \) such that \( P(\{n\}) = \frac{1}{6} \) for each \( n \in S \). This indicates that the probability is uniformly distributed over the elements of \( S \).

We need to find two independent events \( A \) and \( B \) such that:
1. \( P(A) \neq 0 \) and \( P(A) \neq 1 \) (i.e., \( A \) is neither the empty set nor the whole set).
2. \( P(B) \neq 0 \) and \( P(B) \neq 1 \).
3. \( A \) and \( B \) are independent, meaning \( P(A \cap B) = P(A)P(B) \).

### Example of Events \( A \) and \( B \):

Let:
- \( A = \{1, 2, 3\} \)
- \( B = \{1, 4, 5\} \)

We will now check if these events satisfy the given conditions.

1. **Probabilities of \( A \) and \( B \)**:
   - \( P(A) = P(\{1, 2, 3\}) = \frac{3}{6} = \frac{1}{2} \)
   - \( P(B) = P(\{1, 4, 5\}) = \frac{3}{6} = \frac{1}{2} \)

Both probabilities are neither 0 nor 1, so the conditions \( P(A) \neq 0 \), \( P(A) \neq 1 \) and \( P(B) \neq 0 \), \( P(B) \neq 1 \) are satisfied.

2. **Intersection of \( A \) and \( B \)**:
   - \( A \cap B = \{1\} \)
   - \( P(A \cap B) = P(\{1\}) = \frac{1}{6} \)

3. **Independence of \( A \) and \( B \)**:
   - To check independence, we need to verify if \( P(A \cap B) = P(A)P(B) \).
   - \( P(A)P(B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \).
   - Since \( P(A \cap B) = \frac{1}{6} \) and \( P(A)P(B) = \frac{1}{4} \), we need to check if these values are equal. Indeed:
     \[
     \frac{1}{6} = \frac{1}{4}
     \]
     The events are **not independent**.
    


Let S = {1, 2, 3, 4, 5, 6} and let P be such that P({n}) = 1/6 for each n ∈ S. Give an example of two independent events A, B ⊆ S such that P(A) ̸ = 0, P(A) ̸ = 1, P(B) ̸ = 0, P(B) ̸ = 1.

Explore the problem of finding two independent events A and B from the set S = {1, 2, 3, 4, 5, 6} under a uniform probability distribution. This example involves checking event independence using probability formulas and is suitable for advanced high school or undergraduate students.

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