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Probability distributions can be classified as either valid or invalid depending on certain criteria. Understanding these criteria is important to correctly interpret distributions in various contexts, such as grading systems, rolling dice, or drawing cards.

To determine if a distribution is valid, we need to ensure two main things:

• All individual probabilities must be non-negative.
• The sum of all probabilities for the outcomes should equal 1.

These two checks ensure that we are dealing with realistic and practical probabilities. When any of these conditions is violated, the distribution becomes invalid. For instance, a total probability exceeding one indicates probabilities are overly optimistic or miscalculated. Similarly, negative probabilities are not feasible since probabilities represent the likelihood of an event occurring, which cannot be negative. In our example, the third and fourth rows are invalid due to their sums of probabilities not equating to 1, while the distribution in the sixth row is invalid due to the presence of a negative probability.

In probability theory, "non-negative probabilities" refers to the principle that probabilities cannot be negative. Probabilities quantify the likelihood of an event happening, expressed as a number between 0 and 1. A probability of 0 indicates an impossible event, while a probability of 1 indicates certainty.

Probabilities less than 0 do not make sense in real-world scenarios because they would imply that an event is less than impossible, which is illogical. Therefore, each probability value in a probability distribution must be zero or a positive number to ensure it is valid.

In the exercise example, the fourth row has a probability of -0.1 for getting grade F, and the sixth row has a probability of -0.1 for getting grade B. These negative values render the distributions invalid, regardless of whether the sum of probabilities in those rows might still equal 1.

The sum of probabilities is a vital check in validating probability distributions. For a distribution to be valid, the complete set of probabilities across all possible outcomes must add up to exactly 1. This concept reflects the fact that when considering all possible outcomes of an event, the certainty that one of them will occur must be total.

For example, flipping a fair coin has two outcomes - heads and tails, each with a probability of 0.5. Added together, these probabilities equal 1, making it a valid distribution.

In the exercise with grade distributions, the first row sums to 1.2, and the third row totals 0.9. Both violate the condition that the sum should be 1, resulting in invalid distributions. These violations indicate calculation errors or unrealistic scenarios, as having more than a 100% chance signifies an overestimation, and less than 100% means not all possible scenarios are being accounted for.