In probability theory, P(A|B) and P(B|B) refer to conditional probabilities. Here's how to interpret them:

1. P(A|B): This is the conditional probability of event A occurring given that event B has already occurred. It reads as "the probability of A given B." The formula for this is:

   $P(A|B) = \frac{P(A \cap B)}{P(B)}$

   where $P(A \cap B)$ is the probability that both A and B occur, and $P(B)$ is the probability that B occurs.

2. P(B|B): This is the conditional probability of event B occurring given that event B has already occurred. It reads as "the probability of B given B." Since we're given that B has already occurred, this is always 1. Formally:

   $P(B|B) = \frac{P(B \cap B)}{P(B)} = \frac{P(B)}{P(B)} = 1$

In summary:

• P(A|B) is the probability of A occurring when B is known to have occurred.
• P(B|B) is always 1, because if B has already occurred, its conditional probability given itself is certain.

Would you like more details or have any further questions?

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Here are 5 related questions:

1. What is the difference between conditional and unconditional probability?
2. How do you calculate $P(A \cap B)$ in conditional probability?
3. What is the law of total probability and how does it relate to conditional probabilities?
4. What is Bayes' Theorem, and how does it use conditional probability?
5. How does conditional independence affect $P(A|B)$?

Tip: Always check that the denominator $P(B)$ is non-zero when calculating $P(A|B)$, as dividing by zero is undefined.