Define conditional probability. State and prove Baye's theorem.

### Conditional Probability

**Definition:**
Conditional probability is the probability of an event occurring given that another event has already occurred. If \( A \) and \( B \) are two events, the conditional probability of \( A \) given \( B \) is denoted by \( P(A \mid B) \) and is defined as:

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

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

This definition is valid as long as \( P(B) > 0 \).

### Bayes' Theorem

**Statement:**
Bayes' Theorem relates the conditional probability of an event \( A \) given another event \( B \) to the reverse, i.e., the probability of \( B \) given \( A \). It is mathematically stated as:

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

where:
- \( P(A \mid B) \) is the conditional probability of \( A \) given \( B \).
- \( P(B \mid A) \) is the conditional probability of \( B \) given \( A \).
- \( P(A) \) and \( P(B) \) are the probabilities of events \( A \) and \( B \) respectively.

### Proof of Bayes' Theorem

#### Step 1: Start with the definition of conditional probability.
From the definition of conditional probability, we have:

\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad \text{(1)}
\]

Similarly, the conditional probability of \( B \) given \( A \) is:

\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)} \quad \text{(2)}
\]

#### Step 2: Express \( P(A \cap B) \) in two ways.
From equation (1), multiply both sides by \( P(B) \):

\[
P(A \cap B) = P(A \mid B) \cdot P(B) \quad \text{(3)}
\]

From equation (2), multiply both sides by \( P(A) \):

\[
P(A \cap B) = P(B \mid A) \cdot P(A) \quad \text{(4)}
\]

#### Step 3: Equate the two expressions for \( P(A \cap B) \).
Since both equations (3) and (4) represent \( P(A \cap B) \), we can equate them:

\[
P(A \mid B) \cdot P(B) = P(B \mid A) \cdot P(A)
\]

#### Step 4: Solve for \( P(A \mid B) \).
Finally, solve for \( P(A \mid B) \):

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

This equation is Bayes' Theorem.

### Applications of Bayes' Theorem
Bayes' Theorem is widely used in various fields such as medical diagnosis, finance, and machine learning, where it helps in updating the probability of a hypothesis as more evidence or information becomes available.

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Would you like more details or have any questions? Here are five related questions to explore further:

1. How is Bayes' Theorem applied in medical diagnosis to update the probability of a disease given a positive test result?
2. What is the difference between conditional probability and joint probability?
3. How can Bayes' Theorem be extended to more than two events?
4. What is the significance of the prior probability \( P(A) \) in Bayes' Theorem?
5. How is Bayes' Theorem used in Bayesian inference?

**Tip:** When using Bayes' Theorem, ensure that the events involved are well-defined and that the conditional probabilities are correctly interpreted based on the context.

Explore the definition of conditional probability and the proof of Bayes' Theorem with a step-by-step explanation. This page covers key concepts in probability theory, including conditional probability and its application in Bayes' Theorem. Suitable for undergraduate students studying mathematics or statistics.

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