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.