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The question you provided seems to deal with conditional probability and complements of events. Let’s break down the terms first:

- \( A \) and \( B \) are events.
- \( P(A | B) \) refers to the probability of \( A \) occurring given that \( B \) has occurred.
- \( \overline{A} \) refers to the complement of \( A \), meaning \( A \) does not occur.
- The question is asking to select the correct mathematical expression involving \( P(\overline{A} | B) \), the probability of the complement of \( A \) given \( B \).

We know that:
\[
P(A | B) + P(\overline{A} | B) = 1
\]
This means that the probability of \( A \) happening, given \( B \), plus the probability of \( A \) not happening, given \( B \), must sum to 1.

Thus, we can express the complement as:
\[
P(\overline{A} | B) = 1 - P(A | B)
\]

From the answer choices:
- The correct one would be the option that states: \( P(\overline{A} | B) = 1 - P(A | B) \).

Let me know if you'd like a deeper explanation or have any further questions.

Here are five related questions to deepen your understanding:
1. How do you calculate \( P(A | B) \) using Bayes' theorem?
2. What is the difference between independent and mutually exclusive events in probability?
3. How do you compute the joint probability \( P(A \cap B) \)?
4. What is the relationship between conditional probability and total probability?
5. How do complements work in the case of independent events?

**Tip:** Remember that conditional probabilities change when given new information, and their complements can be easily derived using \( 1 - P(A | B) \).

If events A and B are not independent, determine the correct mathematical expression for the complement of P(A | B).

Explore the correct way to express the complement of conditional probabilities, using the complementary rule: P(A | B) + P(\overline{A} | B) = 1. Ideal for Grades 10-12, this concept helps students understand dependencies between events.

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