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When you are determining the probability of an event, why must the probability be between 0 and 1? Explain your answer.
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Short answer: because probability is a measure of how likely an event is, and it is defined as a non‑negative fraction of the total possible outcomes, so it cannot be less than 0 or greater than 1.
More precisely:
- Intuition/frequency view: If you repeat an experiment many times, the probability of an event is the fraction of trials in which it occurs. A fraction cannot be negative and cannot exceed 1 (100%).
- Axiomatic/proof view: A probability P is a measure on the sample space S with P(S) = 1 and P(A) ≥ 0 for every event A. Let A^c be the complement of A. Then
P(A) + P(A^c) = P(S) = 1.
Since P(A^c) ≥ 0, we have P(A) ≤ 1; and by the non‑negativity axiom P(A) ≥ 0. Thus 0 ≤ P(A) ≤ 1.
If any probability were negative or greater than 1, it would contradict these basic definitions and lead to logical impossibilities (for example, the probability of an event plus the probability of its complement would not add to 1).
More precisely:
- Intuition/frequency view: If you repeat an experiment many times, the probability of an event is the fraction of trials in which it occurs. A fraction cannot be negative and cannot exceed 1 (100%).
- Axiomatic/proof view: A probability P is a measure on the sample space S with P(S) = 1 and P(A) ≥ 0 for every event A. Let A^c be the complement of A. Then
P(A) + P(A^c) = P(S) = 1.
Since P(A^c) ≥ 0, we have P(A) ≤ 1; and by the non‑negativity axiom P(A) ≥ 0. Thus 0 ≤ P(A) ≤ 1.
If any probability were negative or greater than 1, it would contradict these basic definitions and lead to logical impossibilities (for example, the probability of an event plus the probability of its complement would not add to 1).
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