Explanation
The question states that the probability of the simultaneous occurrence of n events is equal to the product of their individual probabilities:
P(E1∩E2∩⋯∩En)=P(E1)⋅P(E2)…P(En)
which is written as:
2. Definition of Independence
In statistics, this is the definition of Mutually Independent Events.
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Pairwise Independence: This occurs if P(Ei∩Ej)=P(Ei)P(Ej) for all i=j.
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Mutual Independence: This is a stronger condition where the product rule holds for any sub-collection of the events, including the intersection of all n events.
3. Consequences of the Condition
If this condition holds, it implies:
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The occurrence of any one event does not change the probability of the occurrence of the others.
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The conditional probability P(E1∣E2∩E3…) is simply P(E1).
4. Conclusion
When the joint probability is the product of individual probabilities, the events E1,E2,…,En are Independent Events.
Final Answer:
The events E1,E2,…,En are mutually independent.
