Beliefs depend on the available information. This idea is formalized in probability theory by conditioning. Conditional probabilities, conditional expectations, and conditional probability distributions are treated on three levels: discrete probabilities, probability density functions, and measure theory. Conditioning leads to a non-random result if the condition is completely specified; otherwise, if the condition is left random, the result of conditioning is also random.

Example: A fair coin is tossed 10 times; the random variable X is the number of heads in these 10 tosses, and Y is the number of heads in the first 3 tosses. In spite of the fact that Y emerges before X it may happen that someone knows X but not Y.

Given that X = 1, the conditional probability of the event Y = 0 is

More generally,

One may also treat the conditional probability as a random variable, — a function of the random variable X, namely,

The expectation of this random variable is equal to the (unconditional) probability,

namely,

which is an instance of the law of total probability ${\displaystyle \mathbb {E} (\mathbb {P} (A|X))=\mathbb {P} (A).}$