Statistics and Probability - Conditional probability

Conditional Probability Definition: This refers to the probability of an event $A$ occurring given that event $B$ has already occurred, denoted as $P(A|B)$. Visually, this corresponds to 'shrinking' the sample space; instead of looking at the entire set of possible outcomes, we only consider the outcomes that fall within the circle representing event $B$.

Venn Diagram Interpretation: When looking at a Venn diagram of two events $A$ and $B$, $P(A|B)$ is calculated by taking the area of the intersection ($A \cap B$) and dividing it by the entire area of circle $B$. It ignores any parts of $A$ that are outside of $B$ and ignores the universal set outside of $B$.

Tree Diagrams and Branching: In a tree diagram, conditional probabilities are represented by the second set of branches. If the first branch represents event $A$, the branch following it represents $P(B|A)$. To find the probability of both events happening ($A \cap B$), you multiply the probabilities along the path.

Independent Events: Two events are independent if the occurrence of one does not change the probability of the other. Visually, if $A$ and $B$ are independent, the proportion of $A$ within the $B$ circle is the same as the proportion of $A$ in the whole rectangular sample space. This is expressed as $P(A|B) = P(A)$.

Dependent Events and Sampling: Events are dependent if the outcome of the first event affects the second. This is common in 'without replacement' scenarios. Visually, this is shown on a tree diagram where the denominator of the fraction on the second branch decreases by one because one item has been removed from the total.

Two-Way Tables (Contingency Tables): These tables organize data into rows and columns based on two variables. To find a conditional probability like $P(\text{Category X} | \text{Category Y})$, you restrict your focus to the single row or column representing 'Category Y' and find the proportion of 'Category X' within that specific row or column total.