Why is the probability of one exact value not usually considered for a continuous random variable?
Swiss commemorative stamp of mathematician Jakob Bernoulli, issued 1994, displaying the formula and the graph for the law of large numbers, first proved by Bernoulli in 1713.
For a continuous random variable, it is not meaningful to calculate the probability of the variable taking on one exact value because there are an infinite number of possible values within any given interval on the real number line.1
In the case of continuous variables, probabilities are determined by using a probability density function. Rather than assigning a specific probability to a single point, we calculate the probability that the variable will fall within a specific range. This is achieved by computing the integral of the density function over that interval. Because the area under a single point on a continuous curve is zero, the probability of any exact value is effectively zero, making the focus on intervals necessary for meaningful statistical analysis.213