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 $1 - P(A | B)$.