Conditional probability is the probability of an event occurring given that another event has already occurred. In other words, it measures the likelihood of an event \( A \) happening under the condition that event \( B \) is known to have occurred.

The conditional probability of \( A \) given \( B \), denoted as \( P(A|B) \), is calculated using the formula:

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

Where:
- \( P(A \cap B) \) is the probability that both events \( A \) and \( B \) happen.
- \( P(B) \) is the probability that event \( B \) happens.

### Key points:
1. \( P(A|B) \) is only defined when \( P(B) > 0 \) because the occurrence of \( B \) must be possible.
2. Conditional probability helps refine predictions based on new information. It is widely used in various fields like statistics, machine learning, and everyday reasoning.

### Example:
If you roll two dice, let:
- \( A \) be the event "the sum of the dice is 8."
- \( B \) be the event "the first die shows a 3."

To find \( P(A|B) \), the probability that the sum is 8 given that the first die shows a 3, we calculate:

- There is only one outcome that satisfies both \( A \) and \( B \), which is \( (3, 5) \).
- The probability of event \( B \) (first die is 3) is \( \frac{1}{6} \).

Using the conditional probability formula, we get:

\[
P(A|B) = \frac{1}{36} \div \frac{1}{6} = \frac{1}{6}
\]

Would you like further clarification or details on specific aspects of conditional probability?

### Relative Questions:
1. What are some real-world applications of conditional probability?
2. How does Bayes' theorem relate to conditional probability?
3. Can conditional probability be used to solve problems in machine learning?
4. What is the difference between joint probability and conditional probability?
5. How does conditional probability apply in medical testing (e.g., false positives)?

### Tip:
When dealing with conditional probabilities, always ensure you understand the total probability of the given condition. This helps avoid common pitfalls in interpreting results.

Learn the concept of conditional probability, its formula P(A|B) = P(A ∩ B) / P(B), and explore how it's used to calculate the probability of events given a condition. This lesson provides examples and is ideal for students in Grades 9-12.

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