Conditional Probability

Conditional probability is the probability that an event \( A \) occurs given that another event \( B \) has already occurred.

It is defined as \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \) (

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

), provided that \( P(B) > 0 \).

This formula reflects how new information about one event can affect our belief in another event's occurrence.

For example, suppose we draw a card from a deck and we know it is a red card; because $26$ of the cards in the deck are red (the rest are black,) the probability that the drawn card is a heart is $P(\text{heart} \mid \text{The card is red}) \;$$= \frac{P(\text{heart} \cap \text{The card is red})}{P(\text{The card is red})} \;$$= \frac{P(\text{heart})}{P(\text{The card is red})} \;$$= \frac{13/52}{26/52} \;$$= \frac{13}{26} \;$$= \frac{1}{2}$.

Conditional probability is useful in sequential experiments, decision-making, and updating probabilities based on evidence. It will also help us understanding independence and Bayes' theorem, both of which are crucial for probabilistic reasoning.