Conditional probability is a fundamental concept in probability theory that describes the likelihood of an event occurring, given that another event has already occurred. It's expressed as the probability of event \(A\) happening given that event \(B\) has happened, and it's denoted as \(P(A \mid B)\).

### Definition
The conditional probability of \(A\) given \(B\) is defined by the formula:

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

Here:
- \(P(A \cap B)\) is the probability that both events \(A\) and \(B\) occur.
- \(P(B)\) is the probability that event \(B\) occurs.

This formula assumes that \(P(B) > 0\), as the conditional probability is undefined when \(P(B) = 0\).

### Explanation
- **Numerator (\(P(A \cap B)\))**: This is the probability that both events \(A\) and \(B\) happen simultaneously.
- **Denominator (\(P(B)\))**: This is the probability that event \(B\) happens, serving as the new "total" for the conditional probability calculation.

The formula essentially scales the joint probability \(P(A \cap B)\) by the probability of the given condition \(B\), thus narrowing down the sample space to only those outcomes where \(B\) occurs.

### Example
Imagine a deck of 52 cards. Suppose you want to find the probability of drawing an Ace (event \(A\)) given that the card drawn is a Spade (event \(B\)).

- There are 4 Aces in the deck, and 1 of them is a Spade.
- There are 13 Spades in the deck.

Using the formula:

\[
P(\text{Ace} \mid \text{Spade}) = \frac{P(\text{Ace} \cap \text{Spade})}{P(\text{Spade})} = \frac{\frac{1}{52}}{\frac{13}{52}} = \frac{1}{13}
\]

So, the probability of drawing an Ace given that the card is a Spade is \(\frac{1}{13}\).

### Properties
1. **Multiplication Rule**: The probability of both \(A\) and \(B\) happening can be expressed as:
   
   \[
   P(A \cap B) = P(B) \times P(A \mid B)
   \]

2. **Independence**: If \(A\) and \(B\) are independent events, then \(P(A \mid B) = P(A)\), meaning the occurrence of \(B\) does not affect the probability of \(A\).

Would you like further details or have any questions?

### Follow-up Questions
1. How does conditional probability differ from regular probability?
2. What are some common applications of conditional probability?
3. How can conditional probability be extended to more than two events?
4. How do you calculate conditional probability using Bayes' theorem?
5. Can conditional probability be applied to real-world scenarios like medical testing?

### Tip
When dealing with conditional probabilities, it's essential to carefully define the events and ensure you understand the condition being applied, as this can significantly affect the interpretation of the probability.

Explore the concept of conditional probability, including the key formula P(A | B) = P(A ∩ B) / P(B), with a practical card-drawing example. Learn how to calculate probabilities when given conditions apply, ideal for students in Grades 10-12.

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