Content
Combined Events
Venn diagrams, intersection and union of events, conditional probability, independent events
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Exercises
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Venn diagrams, intersection and union of events, conditional probability, independent events
Want a deeper conceptual understanding? Try our interactive lesson! (Plus Only)
No exercises available for this concept.
A Venn diagram is a visual tool used to illustrate relationships between sets or events. Each event is represented by a circle, and overlaps between circles represent shared outcomes. This helps you clearly see intersections (outcomes common to events), unions (all outcomes in any event), and mutually exclusive events (no overlap).
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Venn diagrams are often filled in with numbers representing the number of samples in each category.
The intersection is the event where both A and B occur simultaneously, denoted A∩B.
The union is the event that at least one of A or B occurs. The union is denoted A∪B and has probability
This formula is sometimes referred to as the inclusion-exclusion rule. It is often rearranged in the form
Events are mutually exclusive if they cannot both occur at once. In this case, the intersection probability is zero:
And therefore
Conditional probability is the probability of event A happening given we already know event B has occurred. It's calculated by taking the probability that both events occur, divided by the probability of the known event B:
Notice that we can rearrange this formula to get a general formula for the probability of multiple events,
Tables of outcomes are useful tools for visualizing interactions between events that are sorted into multiple categories. For example,
(need better table tool)
If two events A and B are independent, then knowing whether or not one happened gives no information on whether or not the other happened, and
Rearranging the conditional probability formula gives the fact that for independent A and B,