Statistics and Probability - Venn Diagrams and Tree Diagrams

Sample Space and Venn Diagrams: The sample space $U$ is represented visually as a rectangular box containing all possible outcomes. Events within this space are depicted as circles. The intersection of two circles, $A \cap B$, represents the region where both events occur simultaneously. The union, $A \cup B$, covers the entire area within both circles, representing the occurrence of either $A$ or $B$ or both.

Complement of an Event: The complement of event $A$, denoted $A'$, consists of all outcomes in the universal set $U$ that are not in $A$. Visually, this is the entire area inside the rectangle but outside the circle representing $A$. The sum of probabilities $P(A) + P(A') = 1$.

Mutually Exclusive Events: These are events that cannot happen at the same time. In a Venn diagram, mutually exclusive events $A$ and $B$ are drawn as two separate, non-overlapping circles. Mathematically, $P(A \cap B) = 0$, meaning their intersection is empty.

Tree Diagrams for Sequential Events: Tree diagrams are used to map out outcomes for multi-stage experiments. Each 'branch' represents a possible outcome, and the probability of that outcome is written along the branch. To find the probability of a specific path (a sequence of events), you multiply the probabilities along those branches. The sum of probabilities for all branches originating from a single node must always equal $1$.

Conditional Probability: This measures the probability of event $A$ occurring given that $B$ has already occurred, denoted $P(A|B)$. Visually in a Venn diagram, this corresponds to restricting your focus entirely to the circle for $B$ and determining what fraction of that circle is occupied by the intersection $A \cap B$.

Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. In a tree diagram, this means the probabilities on the second set of branches remain the same regardless of the outcome of the first set. This is confirmed if $P(A \cap B) = P(A) \times P(B)$.

Combined Events (Addition Rule): The probability that at least one of two events occurs is given by the area of their union. Visually, if you add the areas of circle $A$ and circle $B$, you have counted the overlapping intersection $A \cap B$ twice. Therefore, to find $P(A \cup B)$, you must subtract the intersection once: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.