Derive of the Bayes' theorem using conditional probabilities. Hint: begin with the probability of two events x and y

To derive **Bayes' Theorem** using conditional probabilities, we begin by considering the joint probability of two events, \( A \) and \( B \). The joint probability can be expressed in two different ways based on conditional probabilities.

### Step 1: Define the joint probability

The probability of both events \( A \) and \( B \) happening is called the **joint probability**, denoted as \( P(A \cap B) \), which can be written in terms of conditional probabilities:

\[
P(A \cap B) = P(A \mid B) P(B)
\]
This says the probability of \( A \) and \( B \) happening together is the probability that \( B \) happens multiplied by the probability that \( A \) happens given that \( B \) has already occurred.

Similarly, we can also express the joint probability in the opposite way:

\[
P(A \cap B) = P(B \mid A) P(A)
\]
This states that the probability of both \( A \) and \( B \) happening is the probability that \( A \) happens multiplied by the probability that \( B \) happens given \( A \).

### Step 2: Set the two expressions equal

Since both expressions represent the same joint probability \( P(A \cap B) \), we can equate them:

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

### Step 3: Solve for \( P(A \mid B) \)

To obtain **Bayes' Theorem**, we solve for \( P(A \mid B) \) by dividing both sides of the equation by \( P(B) \) (assuming \( P(B) \neq 0 \)):

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

### Step 4: Interpret the result

This is **Bayes' Theorem**, which allows us to calculate the probability of event \( A \) occurring given that \( B \) has occurred, in terms of the reverse conditional probability \( P(B \mid A) \), the prior probability \( P(A) \), and the marginal probability \( P(B) \).

### Explanation of Terms:
- \( P(A \mid B) \): Posterior probability — the probability of \( A \) given \( B \).
- \( P(B \mid A) \): Likelihood — the probability of \( B \) given \( A \).
- \( P(A) \): Prior probability — the initial probability of \( A \).
- \( P(B) \): Evidence — the total probability of \( B \) across all possible outcomes of \( A \).

This derivation shows how conditional probabilities are related and gives the foundational formula for updating probabilities when new evidence is available.

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Would you like a more detailed explanation of any specific step? 

Here are 5 related questions to expand your understanding:
1. How can Bayes' Theorem be used in real-world applications, such as medical testing?
2. What is the difference between prior and posterior probabilities in the context of Bayes' Theorem?
3. How is the marginal probability \( P(B) \) calculated in practice?
4. What assumptions are made when applying Bayes' Theorem?
5. How does Bayes' Theorem relate to the concept of conditional independence?

**Tip:** Bayes' Theorem is particularly powerful when combining prior beliefs with new evidence, making it fundamental in fields like machine learning and statistical inference.

Derive of the Bayes' theorem using conditional probabilities. Hint: begin with the probability of two events x and y.

Explore a step-by-step derivation of Bayes' Theorem starting from the joint probability of two events. This solution covers key concepts in probability theory, including conditional probabilities and the foundational Bayes' Theorem. Suitable for advanced high school students and early college learners.

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