Uncertainty is a complex and ubiquitous phenomenon. Randomness is an important concept to describe uncertainty, and its quantitative tool is probability. Through an in-depth study of the distribution law of prime numbers, this paper finds that prime number distribution has both randomness and certainty, which is defined as incomplete randomness. The position of prime numbers in the integer sequence is random, but the number of prime numbers in a certain interval is certain. And there are two trend characteristics of prime number distribution. One big trend is that the density of prime numbers gradually decreases; the other trend is that the probability density in the opposite direction increases slightly. Prime number distribution has a certain randomness, and its distribution is completely controlled by natural laws. There is no accidental cover-up and interference caused by minor factors. The number of prime numbers is fixed. Although there is no accurate function expression, it has a certain degree of certainty. This special type of distribution presents a fixed result and is an incomplete random distribution. The total probability of a particular event is calculated as the cumulative probability: P (total) = ∑P (n). The total cumulative probability as a quantitative tool for incomplete randomness is a new concept. Unlike classical probability, its value is allowed to be greater than the constant 1.The discovery of incomplete randomness helps to find the law of prime number distribution, deepen the understanding of the laws of the universe, and broaden the deeper thinking about the nature of nature.Many conjectures involving prime numbers are unresolved problems, some of which have been around for 300 years. Incomplete randomness can provide a new and unique perspective. This article applies the incomplete random distribution theorem and attempts to give proofs of some of these problems, such as the Mersenne prime conjecture and the Collatz conjecture.