Conditional Probability

$$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$

the conditional probability, which describes the probability of an event occurring given that another event has already occurred.

The conditional probability of event $A$ given event $B$ is denoted as $P(A|B)$

• Example: $P(J=true|W=true)$ represents the probability of event $J$ being true, given that event $W$ is true.

Mathematical Definition

The conditional probability is given by the formula:

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

$P(A\cap B)$ is the joint probability of A and $B$, and $P(B)$ is the probability of event $B$.

• Key Characteristics:

  • Focuses on the relationship between two events.
  • The occurrence of one event affects the likelihood of the other.
  • Only defined when $P(B)>0$

• Bayesian Approach: Conditional probability plays a central role in Bayesian statistics. Using Bayes’ Theorem, we update our beliefs about the likelihood of events based on new evidence: