Foundations of probability: within standard probability theory, valid probabilities for any event must lie within what numerical range?

Introduction / Context:
Probability quantifies uncertainty on a normalized 0 to 1 scale. Understanding this numeric range is foundational for statistics, risk analysis, and decision-making. It ensures calculations, comparisons, and interpretations (like odds, likelihoods, and confidence) remain consistent and meaningful across contexts such as quality control, forecasting, and simulation studies.

Given Data / Assumptions:

• A probability of 0 means the event cannot occur.
• A probability of 1 means the event is certain.
• All other events fall between 0 and 1 inclusive.

Concept / Approach:
By definition, probability P(E) satisfies 0 ≤ P(E) ≤ 1. Percentages (0% to 100%) are an alternative expression but must be converted back to the 0–1 scale for calculations. Values outside this interval indicate a modeling or computational error. Normalization on [0, 1] allows additive and complementary relationships such as P(E) + P(not E) = 1 to hold consistently across analyses.

Step-by-Step Solution:

Verification / Alternative check:
Any valid probability mass or density function integrates/sums to 1 over its sample space; individual event probabilities cannot exceed 1 or fall below 0.

Why Other Options Are Wrong:

• 0.00 to 0.10: too narrow; excludes many legitimate probabilities.
• 0 to 100: looks like percentages, but the unit is incorrect in probability notation.
• All/None: incorrect since one correct numeric range exists.

Common Pitfalls:
Mixing percentages with decimal probabilities without conversion; reporting probabilities greater than 1 due to double counting.

Final Answer:
0.00 to 1.00