1.5.1 Axiom of Nonnegativity

This axiom states that the probability of any event occurring is always equal to or greater than zero. Therefore, if an event  has some probability of occurring, it should be greater than zero. Conversely, if  is impossible, its probability of occurring should be zero, not negative.

        

There is no such thing as negative probability. This is logical because when we discuss an impossible event, we never assign a negative probability, such as a negative 25% chance of rain. Instead, we simply state that there is a zero percent chance or no chance of rain.

Example 1.24. When we talk about a standard die, what is the probability of an event like "rolling the number seven"? Even though rolling a seven is impossible since a die only has numbers from 1 to 6, its probability cannot be less than zero or negative. It will always be zero, represented as:

        

Note that this axiom does not impose any restrictions on the upper limit of probability. It just rules out the existence of negative probabilities.