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In probability theory, a probability distribution adheres to specific rules that dictate how probabilities are assigned to events. These rules help ensure that every probability model is logical and consistent.

One of the most important elements is that the probability of any event is a numerical value between 0 and 1.
This reflects how likely an event is to occur, with 0 representing an impossible event and 1 a certainty.

• A probability of 0 means the event will never happen. For example, the probability of rolling a 7 on a standard 6-sided die is 0.
• A probability of 1 indicates certain occurrence. Rolling a number between 1 and 6 on a 6-sided die has a probability of 1 because one of those numbers is guaranteed to show up.

Understanding probability rules helps in evaluating models and theories in real-world situations, ensuring predictions are realistic and accurate.

Nonnegative probabilities are a cornerstone of probability distributions.
Simply put, they refer to probabilities that cannot be less than 0.
In practical terms, this means you can't have a negative chance of something happening.
These probabilities must always be between 0 and 1, where fractions or decimal forms of percentages are used to express likelihood.

This property is crucial because it aligns with our intuitive understanding of probability.
Seeing as probabilities express possible outcomes occurring, it would be nonsensical to represent an event occurring with a negative number.
For example:

• The phrase "There's a -25% chance of rain" doesn't make logical sense, just as there couldn't be a negative quantity of occurrences in our understanding of reality.

Rather, probabilities depict the scale of possibility from nonexistent (0) to certainty (1).

The principle that the sum of probabilities equals one is another fundamental aspect of probability distributions.
When dealing with multiple possible outcomes of a random event, the probabilities of all individual outcomes combined must result to 1.

This total reflects the certitude that one of these possible outcomes must occur.
Mathematically, this is shown as: \[\sum P(x) = 1\]For example:

• If you flip a coin, there are two possible outcomes: heads or tails. If both have an equally likely chance of occurring, each has a probability of 0.5. Added together, these probabilities sum to 1, affirming that one outcome will indeed occur.

This rule ensures a complete account of all possibilities, confirming that the distribution is valid and reflective of real scenarios.