The probability of an event quantifies the likelihood of that event occurring and is expressed as a number between 0 and 1 for several important reasons:
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Definition of Probability: By definition, the probability \( P(A) \) of an event \( A \) is a value that represents the ratio of the number of favorable outcomes to the total number of possible outcomes. This ratio cannot exceed 1 (which would imply that there are more favorable outcomes than total outcomes), and it cannot be less than 0 (which would suggest a negative likelihood).
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Interpretation of Outcomes: A probability of 0 means that the event cannot occur (it is impossible), while a probability of 1 means that the event is certain to occur (it is certain). Any probability value in between (e.g., 0.25, 0.5, 0.75) reflects varying levels of likelihood of the event happening.
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Normalization: In any probability space, the sum of the probabilities of all possible outcomes must equal 1. This is known as the normalization condition. If any probability exceeded 1 or was below 0, it would disrupt this balance and contradict the foundational rules of probability.
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Logical Consistency: Allowing probabilities below 0 or above 1 would lead to paradoxes and contradictions in probability theory and statistics. For example, if we had a probability of -0.2 or 1.5 for certain events, it wouldn't make sense when considering the complete picture of outcomes.
In summary, constraining probabilities between 0 and 1 ensures that they are interpretable, mathematically consistent, and aligned with our understanding of events and their occurrences in probability theory.
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