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.