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In probability theory, it's crucial to understand what makes probabilities valid. Valid probabilities must meet two conditions:

• Each probability must be between 0 and 1. This means no probability can be negative or exceed 1. Probabilities are essentially measurements of likelihood and, as such, cannot logically be less than 0 (impossible event) or more than 1 (certain event).
• The sum of all probabilities for each possible outcome must equal 1. This requirement arises because the totality of all possible outcomes in any given probability experiment must account for 100% of the probability space.

Let's consider the scenario where probabilities are assigned to four events. For these assignments to be valid, each probability should fall within the 0 to 1 range, and the coupling of all these probabilities should collectively sum to 1. If any of these conditions isn't met, then the probability set is not valid, as seen in the problem given. Here, while each individual probability is within the permissible range, the total sum deviates, leading to invalidity.

Subjective probability is a concept that arises when the probabilities of certain events are not derived from historical data or frequentist interpretations but rather from an individual's personal judgment or experience.
In other words, subjective probabilities reflect a decision maker’s belief about the likelihood of various outcomes. This belief can be influenced by knowledge, perception, or even intuition, which may introduce biases but also consider unique insights into probabilities.
Consider the same experiment with four subjectively assigned probabilities. While such assignments reflect the decision maker’s personal assessment, they still need to adhere to the fundamental principles of valid probabilities. This means even subjective assessments should ensure individual probabilities are between 0 and 1 and collectively add up to 1. Despite the subjective nature, the mathematical framework remains non-negotiable in maintaining the consistency and validity of these probability assignments.

One of the cardinal rules in probability is the sum of probabilities for all possible outcomes must be exactly 1.

• This rule ensures that all potential events are accounted for and exhaustively cover the entire probability space.
• Any deviation from this total can suggest missing a potential outcome or an error in estimating the involved probabilities.

In practical applications such as the described exercise, the sum of given probabilities was notably 0.85. This sum should have been 1 for the probability assignments to be deemed valid.
Such discrepancies require careful reevaluation to identify where an error may have occurred. It could be due to incomplete enumeration of possible outcomes or inaccurate estimations. Ensuring the total probability sums to 1 underlines the internally consistent nature of probability theory, affirming that every outcome and its likelihood are thoroughly mapped.