Given the three events $A$, $B$, and $C$ with $P(C) > 0$, let's analyze each statement to determine which are correct.

1. $P(A^c \mid C) = 1 - P(A \mid C)$

• True: This statement is correct. The conditional probability of the complement of $A$ given $C$ is equal to $1$ minus the conditional probability of $A$ given $C$. Mathematically, this can be written as: $P(A^c \mid C) = 1 - P(A \mid C)$ This follows directly from the definition of conditional probability and the fact that $P(A) + P(A^c) = 1$.

2. $P(\emptyset \mid C) = 0$

• True: This statement is correct. The probability of the empty set given any event $C$ is always 0. This is because the empty set $\emptyset$ represents an impossible event, so $P(\emptyset \mid C) = 0$.

3. $P(A \mid C) \leq 1$

• True: This statement is correct. The probability of any event $A$ conditioned on $C$ must lie between 0 and 1, inclusive. Therefore, $P(A \mid C) \leq 1$.

4. If $A \subset B$ then $P(A \mid C) \leq P(B \mid C)$

• True: This statement is correct. If $A \subset B$, then $P(A \mid C)$ is less than or equal to $P(B \mid C)$. This is because the probability of a subset (smaller event) occurring is less than or equal to the probability of the larger event occurring when conditioned on the same event $C$.

Conclusion:

• All four statements are correct.

Would you like more details or have any questions?

Here are some related questions:

1. What is the formula for conditional probability?
2. Can you prove the property $P(A \mid C) + P(A^c \mid C) = 1$ using basic probability rules?
3. What is the interpretation of $P(A \mid C)$ in real-world scenarios?
4. How does the law of total probability apply when conditioning on an event $C$?
5. Can you provide examples where $P(A \mid C) = P(A)$, meaning event $C$ is independent of event $A$?

Tip: Remember that the conditional probability formula $P(A \mid C) = \frac{P(A \cap C)}{P(C)}$ is only valid when $P(C) > 0$.