Mutually Exclusive and Overlapping Probabilities
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Introduction
In probability theory, events can be classified as mutually exclusive or overlapping. Understanding these concepts is crucial for solving probability problems.
1. Mutually Exclusive Events
Two events are mutually exclusive if they cannot happen at the same time. In other words, if one event occurs, the other cannot.
Formula:
\( P(A \text{ or } B) = P(A) + P(B) \)
2. Overlapping Events
Events that are not mutually exclusive and can occur simultaneously are termed as overlapping events.
Formula:
\( P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) \)
Steps to Solve Problems:
- Identify if the given events are mutually exclusive or overlapping.
- If the events are mutually exclusive, use the formula for mutually exclusive events.
- If the events are overlapping, use the formula for overlapping events.
- Substitute the given probabilities into the formula.
- Calculate the required probability.
Example
Two events are said to be mutually exclusive if they cannot happen at the same time.
Example: Consider rolling a standard six-sided die. Let event \( A \) be rolling an even number and event \( B \) be rolling a number less than 3.
\( P(A \text{ and } B) \) = \( P(2) \) = \( \frac{1}{6} \).
Overlapping Events
Two events are overlapping if they can happen at the same time.
Example: Consider drawing a card from a standard deck. Let event \( C \) be drawing a red card and event \( D \) be drawing a heart.
Since all hearts are red cards, \( P(C \text{ and } D) \) = \( P(D) \) = \( \frac{1}{4} \).
