The total probability of all possible values of a random variable is always 1. This means that if you sum (or integrate, in the continuous case) the probabilities of every possible outcome that the random variable can take, the result is 1.

• For a discrete random variable with possible values x1,x2,…x_1,x_2,\ldots x1​,x2​,…, the sum of the probabilities is

∑iP(X=xi)=1.\sum_i P(X=x_i)=1.i∑​P(X=xi​)=1.

• For a continuous random variable with probability density function fX(x)f_X(x)fX​(x), the integral over all possible values is

∫−∞∞fX(x) dx=1.\int_{-\infty}^{\infty}f_X(x),dx=1.∫−∞∞​fX​(x)dx=1.

This is a fundamental property of probability distributions ensuring that the random variable takes some value in its domain with certainty

. The law of total probability further relates probabilities of events by partitioning the sample space into mutually exclusive and exhaustive parts, but the total probability over all possible values remains 1