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How does a probability mass function assign probabilities to discrete random-variable values?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In probability theory, a discrete random variable is one that assumes a finite or countably infinite sequence of values. The probability mass function (PMF), denoted as $f (x)$ , is the mathematical tool used to assign probabilities to these specific values.1

The PMF functions by providing the exact probability for each individual outcome of the random variable. To be valid, the function must satisfy two fundamental requirements:

Non-negativity: The value of $f (x)$ must be greater than or equal to zero for every possible value of the random variable.1
Normalization: The sum of the probabilities assigned to all possible values of the random variable must equal exactly one.1

Unlike a probability density function, which is used for continuous variables and represents probability over intervals rather than at specific points, the PMF directly defines the probability distribution for discrete outcomes. These values are then used to calculate statistical measures such as the expected value and variance.1