Sample space in the context of "Trial and error"

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space).

For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.5 for X = tails (assuming that the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values.

Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion).Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample. Probability density is the probability per unit length, in other words. While the absolute likelihood for a continuous random variable to take on any particular value is zero, given there is an infinite set of possible values to begin with. Therefore, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample.

More precisely, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of a continuous variable's PDF over that range, where the integral is the nonnegative area under the density function between the lowest and greatest values of the range. The PDF is nonnegative everywhere, and the area under the entire curve is equal to one, such that the probability of the random variable falling within the set of possible values is 100%.

Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see § Events in probability spaces, below).

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers to neither randomness nor variability but instead is a mathematical function in which

• the domain is the set of possible outcomes in a sample space (e.g. the set ${\displaystyle \{H,T\}}$ which are the possible upper sides of a flipped coin heads ${\displaystyle H}$ or tails ${\displaystyle T}$ as the result from tossing a coin); and
• the range is a measurable space (e.g. corresponding to the domain above, the range might be the set ${\displaystyle \{-1,1\}}$ if say heads ${\displaystyle H}$ mapped to -1 and ${\displaystyle T}$ mapped to 1). Typically, the range of a random variable is a subset of the real numbers.

Informally, randomness typically represents some fundamental element of chance, such as in the roll of a die; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup.

In probability theory and statistics, the empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, i.e. by means not of a theoretical sample space but of an actual experiment. More generally, empirical probability estimates probabilities from experience and observation.

Given an event A in a sample space, the relative frequency of A is the ratio ⁠${\displaystyle {\tfrac {m}{n}},}$⁠ m being the number of outcomes in which the event A occurs, and n being the total number of outcomes of the experiment.

In probability theory, an experiment or trial (see below) is the mathematical model of any procedure that can be infinitely repeated and has a well-defined set of possible outcomes, known as the sample space. An experiment is said to be random if it has more than one possible outcome, and deterministic if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as a Bernoulli trial.

When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number of events, all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess the empirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods of statistical analysis.

In probability theory, an outcome is a possible result of an experiment or trial. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a sample space.