The Addition Rule and Complements
Learn how to calculate probabilities for "OR" events, understand mutual exclusivity, and use the complement rule to simplify complex problems.
Disjoint Events (Mutually Exclusive)
Definition
Two events are disjoint (or mutually exclusive) if they cannot occur at the same time. They have no outcomes in common.
Disjoint Example
Rolling a single die:
• Event A: Rolling an even number (2, 4, 6)
• Event B: Rolling a 5
You cannot roll both at the same time.
Not Disjoint Example
Rolling a single die:
• Event A: Rolling an even number (2, 4, 6)
• Event B: Rolling a number > 3 (4, 5, 6)
Overlap: {4, 6} are in both.
The General Addition Rule
To find the probability of Event A OR Event B occurring, we add their individual probabilities but subtract the overlap to avoid double-counting.
lap
The Double-Counting Error
When you add and , the overlapping region (intersection) is counted twice. We subtract once to correct this.
InteractiveVenn Diagram Explorer
The Complement Rule
Definitions
The complement of event E (denoted or ) consists of all outcomes in the sample space that are NOT in E.
LogicLens Strategy: "At Least One"
Calculating the probability of "at least one" success can be difficult directly. It's often easier to use the complement:
Example: If there is a 20% chance of rain each day, calculating "rain at least once in a week" directly is complex. Calculating "no rain all week" is simpler, then just subtract from 1.
Adaptive Assessment
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