In probability theory, what is the numerical value of the probability of an impossible event?

Introduction / Context:
This is a conceptual question about the basic axioms of probability. An impossible event is one that can never occur under the given experimental setup, no matter how many times the experiment is repeated. Understanding the correct probability value for such an event is fundamental to mastering probability theory and interpreting probability values correctly.

Given Data / Assumptions:

We are working within standard probability theory.

An event is called impossible if it has no possible outcomes in the sample space.

We must choose the correct numerical value for the probability of an impossible event from the given options.

Concept / Approach:
By the axioms of probability, the probability of any event E satisfies 0 ≤ P(E) ≤ 1. The probability of the sure (certain) event is 1, and the probability of the empty event, or impossible event, must be 0. The empty event has no favourable outcomes, so the ratio of favourable outcomes to total outcomes is 0 divided by the total, which is 0. This is built into every standard definition of a probability measure.

Step-by-Step Reasoning:
Step 1: Consider an experiment with a finite sample space S and let n(S) represent the number of all possible outcomes.Step 2: For a given event E, let n(E) be the number of favourable outcomes for that event.Step 3: The basic definition of probability in this finite, equally likely case is P(E) = n(E) / n(S).Step 4: For an impossible event, E is the empty set, and hence n(E) = 0.Step 5: Therefore, P(E) = 0 / n(S) = 0.Step 6: This aligns with the general axiom that the probability of the empty set is always 0.

Verification / Alternative check:
In measure theoretic terms, probability is a function P defined on events such that P(S) = 1 and P is countably additive. The empty set has measure zero in all standard probability spaces because it cannot be decomposed into any non empty events and carries no weight in sums. Any other value, such as a negative number or a positive number, would contradict the property that probabilities must lie between 0 and 1 and that P(S) = 1 while P of disjoint events adds to the total.

Why Other Options Are Wrong:
The value 1 is the probability of a certain event, not an impossible event. A negative value such as -1 is not allowed because probabilities cannot be less than 0. The value 0.1 or 0.5 would imply that the supposedly impossible event actually has some chance of occurring, which contradicts the definition. Only 0 is consistent with both the intuitive and formal definitions of an impossible event.

Common Pitfalls:
Some learners confuse an impossible event with a very unlikely event. An event with very small probability, such as 0.001, is unlikely but still possible; an impossible event truly cannot happen and therefore must have probability 0. Always distinguish between improbable events (small but positive probabilities) and impossible events (exactly 0 probability).

Final Answer:
The probability of an impossible event is 0.