Explain the concept of conditional probability with the help of a suitable example.

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as \( P(A|B) \), which reads as "the probability of A given B."

The formula for conditional probability is:

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

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

### Example

Let's consider an example involving a deck of cards. Suppose we want to find the probability of drawing an Ace (event A) given that we have drawn a card from the Hearts suit (event B).

1. **Total number of cards in a deck**: 52
2. **Number of Aces in the deck**: 4 (one in each suit)
3. **Number of Hearts in the deck**: 13 (since there are 13 cards in each suit)
4. **Number of Aces in Hearts**: 1 (the Ace of Hearts)

To find \( P(A|B) \):

- **\( P(A \cap B) \)**: The probability of drawing the Ace of Hearts is the probability of drawing one specific card out of 52, which is \( \frac{1}{52} \).

- **\( P(B) \)**: The probability of drawing any card from the Hearts suit is \( \frac{13}{52} \) because there are 13 Hearts in the deck.

Using the conditional probability formula:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{52}}{\frac{13}{52}} = \frac{1}{13} \]

Therefore, the probability of drawing an Ace given that it is from the Hearts suit is \( \frac{1}{13} \).

### Summary
Conditional probability provides a way to update the likelihood of an event based on new information. In this example, knowing that the card is from the Hearts suit changes the probability of drawing an Ace.

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### Further Questions
1. What is the significance of conditional probability in real-world applications?
2. How does Bayes' theorem relate to conditional probability?
3. Can you provide another example involving conditional probability?
4. How do you calculate the probability of independent events?
5. What is the difference between conditional and joint probability?
6. How does conditional probability apply to medical testing?
7. Can you explain the concept of the law of total probability?
8. What are some common mistakes to avoid when calculating conditional probability?

### Tip
When dealing with conditional probabilities, always ensure that the condition (event B) has a non-zero probability, as dividing by zero would make the calculation invalid.

Explore conditional probability with examples, including the probability of drawing an Ace given it is from the Hearts suit. Learn the formula and its application in probability theory. Suitable for Grades 10-12.

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