1. Conditional Probability

Conditional probability measures the probability of an event A occurring given that another event B has already occurred.

• $P(A|B)$ = probability of A given B
• $P(A \cap B)$ = probability that both A and B occur
• $P(B)$ = probability of B

Key Points

• Conditional probability changes the sample space to B.
• If A and B are independent, then $$P(A|B) = P(A)$$

Example:

• A card is drawn from a deck of 52 cards.
• Let $A$ = "card is a King" ($4/52$)
• Let $B$ = "card is a face card" ($12/52$)

2. Multiplication Rule

Conditional probability leads to the general multiplication rule:

• Useful for dependent events.
• If A and B are independent, $P(A \cap B) = P(A) \cdot P(B)$.

3. Bayes’ Theorem

Bayes’ Theorem allows us to update the probability of an event based on new information.

Suppose $E_1, E_2, ..., E_n$ are mutually exclusive and exhaustive events (partition of sample space) and A is an event. Then:

• $P(E_i)$ = prior probability of $E_i$
• $P(A|E_i)$ = probability of observing A given $E_i$
• $P(E_i|A)$ = posterior probability of $E_i$ after observing A

Steps to Use Bayes’ Theorem

1. Identify the partition events $E_1, E_2, ..., E_n$.
2. Determine the prior probabilities $P(E_i)$.
3. Determine likelihoods $P(A|E_i)$.
4. Apply formula to find posterior probability $P(E_i|A)$.

Example: Medical Test

• Disease prevalence: $P(D) = 0.01$
• Test positive if diseased: $P(T+|D) = 0.99$
• Test positive if healthy: $P(T+|D') = 0.05$

Probability that a person has the disease given a positive test:

• Even with a positive test, probability of disease ≈ 16.6%
• Shows importance of prior probability (prevalence).

4. Summary Table

Key Points to Remember

1. Conditional probability adjusts probabilities based on known events.
2. Multiplication rule links joint probability to conditional probability.
3. Bayes’ theorem is widely used in diagnostics, risk analysis, and machine learning.