Identify which two statements must be true for independent events A and B.

1. Understanding Independence: Two events, $A$ and $B$, are independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is defined by the condition $P(A \cap B) = P(A)P(B)$.

2. Analyzing Conditional Probability: The conditional probability of event $A$ given event $B$ is denoted as $P(A|B)$ and is defined as $P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) \
eq 0$. Similarly, the conditional probability of event $B$ given event $A$ is $P(B|A) = \frac{P(A \cap B)}{P(A)}$, provided $P(A) \
eq 0$.

3. Applying Independence to Conditional Probability: For independent events, we can substitute $P(A \cap B) = P(A)P(B)$ into the conditional probability formulas:
- $P(A|B) = \frac{P(A)P(B)}{P(B)} = P(A)$ (if $P(B) \
eq 0$)
- $P(B|A) = \frac{P(A)P(B)}{P(A)} = P(B)$ (if $P(A) \
eq 0$)

4. Evaluating the Statements: Let's examine each statement based on the definition of independence and the derived conditional probabilities:
- "The probability of $A$ is not the same as the probability of $B$." This is not necessarily true. Independent events can have equal probabilities (e.g., flipping two fair coins, where the probability of heads on the first is 0.5 and on the second is 0.5).
- "The conditional probability of $A$ given $B$ is the same as the probability of $A$." Based on step 3, $P(A|B) = P(A)$. This statement must be true for independent events (assuming $P(B) \
eq 0$, which is usually implied in such problems).
- "The conditional probability of $B$ given $A$ is the same as the probability of $B$." Based on step 3, $P(B|A) = P(B)$. This statement must also be true for independent events (assuming $P(A) \
eq 0$).
- "The conditional probability of $A$ given $B$ is not the same as the conditional probability of $B$ given $A$." This is not necessarily true. If $P(A) = P(B)$, then $P(A|B) = P(A) = P(B) = P(B|A)$.

1. Identifying the Two True Statements: The two statements that must be true for independent events $A$ and $B$ are: "The conditional probability of $A$ given $B$ is the same as the probability of $A$." and "The conditional probability of $B$ given $A$ is the same as the probability of $B$."