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.

Learn how to interpret conditional probabilities P(A|B) and P(B|B) in probability theory. This detailed guide covers key formulas and concepts for students in Grades 10-12, including how to calculate conditional probabilities and the certainty of P(B|B).

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