Last Updated on December 12, 2025 by Rajeev Bagra

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In probability theory, there is a surprising but elegant fact:

This means an event can be independent of itself only if its probability is either 0 (impossible) or 1 (certain).
This article explains why, and shows practical business scenarios where such events naturally occur.

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Why Only Probability 0 or 1?

An event (A) is independent of itself when:

Since the intersection of an event with itself is just the same event:

the equation becomes:

Solving gives:

Therefore, the only solutions are:

If (0<P(A)<1), then:

meaning the event cannot be independent of itself.

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Understanding the Intuition

• If an event always happens, knowing it happened once does not change its probability — it was already certain.
• If an event never happens, no amount of information changes the fact that it is impossible.
• If an event has probability between 0 and 1, learning that it already occurred makes the probability jump to 1, so independence fails.

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Real Business Examples Where Events Are Independent of Themselves

Here are real-world business scenarios where events have probability 0 or 1, making them independent of themselves.

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1. Annual Corporate Tax Filing Deadline (Probability = 1)

The legally fixed filing deadline arrives every year without exception.

This certainty makes it self-independent.

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2. Discontinued Product Generates Sales (Probability = 0)

If a product is permanently discontinued, it cannot generate sales.

An impossible event is self-independent by definition.

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3. Office Closed Every Sunday (Probability = 1)

If company policy states the office is always closed on Sunday:

This is a guaranteed weekly event.

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4. Dissolved Company Makes a Bank Transaction (Probability = 0)

A legally dissolved entity cannot execute new transactions.

Again: impossible → probability 0 → self-independent.

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5. Automated Daily Server Backup Completes (Probability = 1)

Assuming reliable automation, the daily backup always runs at midnight.

Certainty leads to self-independence.

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6. Mandatory Quarterly Compliance Audit Occurs (Probability = 1)

Regulated industries require quarterly audits without exception.

Another example of a guaranteed event.

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Summary Table

Business Event	Probability	Type
Tax filing deadline occurs	1	Certain event
Discontinued product sale	0	Impossible event
Office closed on Sunday	1	Certain event
Dissolved company transaction	0	Impossible event
Daily server backup runs	1	Certain event
Quarterly compliance audit	1	Certain event

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Final Thought

Events with probability 0 or 1 behave differently from everyday uncertain events. They are either guaranteed or impossible, and therefore cannot provide new information that changes their own probability. This is why they are the only events considered independent of themselves.