When you are determining the probability of an event, why must the probability be between 0 and 1? Explain your answer.

Question

helpp

Answer 1

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:

1. 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).

2. 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.

3. 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.

4. 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.