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How do discrete random variables differ from continuous random variables?

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

In probability theory, a random variable is a numerical description of an experiment's outcome. The primary difference between discrete and continuous random variables lies in the values they can assume and how their probabilities are calculated.12

Discrete Random Variables

A discrete random variable assumes a finite number or an infinite sequence of values. Its probability distribution is defined by a probability mass function, which assigns a specific probability to each possible value. For these variables, the sum of all individual probabilities must equal 1.21

Continuous Random Variables

A continuous random variable can assume any value within an interval on the real number line. Because there are infinitely many values in any interval, the probability of the variable taking on any exact, specific value is zero. Instead, probabilities are determined using a probability density function, where the probability of the variable falling within a specific range is found by calculating the integral of the function over that interval. The total area under the density function must equal 1.21