What is the probability of the union of mutually exclusive events?
Understand the Problem
The question is asking about the probability of the union of mutually exclusive events, which refers to the likelihood of either of the events happening when these events cannot occur at the same time. The key idea here is to understand how to calculate the total probability through basic probability rules for such events.
Answer
The probability of the union of mutually exclusive events is given by $$ P(A_1 \cup A_2 \cup \ldots \cup A_n) = P(A_1) + P(A_2) + \ldots + P(A_n) $$
Answer for screen readers
The probability of the union of mutually exclusive events ( A_1, A_2, \ldots, A_n ) is given by:
$$ P(A_1 \cup A_2 \cup \ldots \cup A_n) = P(A_1) + P(A_2) + \ldots + P(A_n) $$
Steps to Solve
- Identify the Events and Their Probabilities
Let's denote the mutually exclusive events as ( A_1, A_2, \ldots, A_n ) with their respective probabilities ( P(A_1), P(A_2), \ldots, P(A_n) ).
- Understand the Formula for Union of Mutually Exclusive Events
When events are mutually exclusive, the probability of the union can be calculated using the formula:
$$ P(A_1 \cup A_2 \cup \ldots \cup A_n) = P(A_1) + P(A_2) + \ldots + P(A_n) $$
- Add Up the Probabilities
To find the total probability of the union, simply add the individual probabilities of the events:
$$ P(A_1 \cup A_2) = P(A_1) + P(A_2) $$
For ( n ) events, it would look like:
$$ P(A_1 \cup A_2 \cup A_3) = P(A_1) + P(A_2) + P(A_3) $$
- Conclusion
The final result gives you the probability that at least one of the events occurs, knowing that they cannot occur at the same time.
The probability of the union of mutually exclusive events ( A_1, A_2, \ldots, A_n ) is given by:
$$ P(A_1 \cup A_2 \cup \ldots \cup A_n) = P(A_1) + P(A_2) + \ldots + P(A_n) $$
More Information
The concept of mutually exclusive events is crucial because it simplifies the calculation of probabilities. If events cannot happen simultaneously, it means the occurrence of one event completely rules out the possibility of the others happening at the same time. This property can significantly affect probability calculations in real-world scenarios.
Tips
- Ignoring Exclusivity: A common mistake is to apply the addition rule for non-mutually exclusive events. Make sure the events are mutually exclusive before using the simplified formula.
- Overlapping Events: Mixing events that can occur at the same time with those that cannot will lead to incorrect calculations.
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