When one tosses a coin one is not sure about the outcome. The outcome is random. But one knows it is either a Head or a Tail. Again one knows something more too. One knows nearly there is 50% chance that Head will turn up and 50% chance that Tail will turn up. When one tosses two coins the number of Heads could be 0, 1 and 2 with chances 25%, 50% and 25%. this is called Distribution and knowing the distribution helps in making decisions. Statistics is the science of making decisions in the face of uncertainty.

The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.

Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input, such as from random number generators or pseudorandom number generators, are important techniques in science, particularly in the field of computational science. By analogy, quasi-Monte Carlo methods use quasi-random number generators.

According to Ramsey theory, ideal randomness is impossible especially for large structures. For example, professor Theodore Motzkin pointed out that "while disorder is more probable in general, complete disorder is impossible". Misunderstanding of this can lead to numerous conspiracy theories.

The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling, but later in connection with physics. Statistics is used to infer the underlying probability distribution of a collection of empirical observations. For the purposes of simulation, it is necessary to have a large supply of random numbers—or means to generate them on demand.

Algorithmic information theory studies, among other topics, what constitutes a random sequence. The central idea is that a string of bits is random if and only if it is shorter than any computer program that can produce that string (Kolmogorov randomness), which means that random strings are those that cannot be compressed. Pioneers of this field include Andrey Kolmogorov and his student Per Martin-Löf, Ray Solomonoff, and Gregory Chaitin. For the notion of infinite sequence, one normally uses Per Martin-Löf's definition. That is, an infinite sequence is random if and only if it withstands all recursively enumerable null sets. The other notions of random sequences include, among others, recursive randomness and Schnorr randomness, which are based on recursively computable martingales. It was shown by Yongge Wang that these randomness notions are generally different.

Randomness occurs in numbers such as log(2) and pi. The decimal digits of pi constitute an infinite sequence and "never repeat in a cyclical fashion." Numbers like pi are also considered likely to be normal, which means their digits are random in a certain statistical sense.

Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases.

In statistics, randomness is commonly used to create simple random samples. This allows surveys of completely random groups of people to provide realistic data that is reflective of the population. Common methods of doing this include drawing names out of a hat, or using a random digit chart (a large table of random digits).

Question I: How do you create an event with 20% chance using dice rolls?

Question II: What is the minimum size of a group of people where there will always be three people who know each other or there will be three people who do not know each other?