Sample space
In probability theory, the sample space (also called sample description space or possibility space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, are listed as elements in the set. It is common to refer to a sample space by the labels "S", Ω, or "U" (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite . A subset of the sample space is an event, denoted by formula_1. If the outcome of an experiment is included in formula_1, then event formula_1 has occurred. For example, if the experiment is tossing a single coin, the sample space is the set formula_4, where the outcome formula_5 means that the coin was heads and the outcome formula_6 means that the coin was tails. The possible events are formula_7 and formula_8. For tossing two coins, the sample space is formula_9, where the outcome is formula_10 if both coins are heads, formula_11 if the first coin is heads and the second  is tails, formula_12 if the first coin is tails and the second is heads, and formula_13 if both coins are tails. For tossing a single six-sided die, where the result of interest is the number of pips facing up, the sample space is formula_14. A well-defined, non-empty sample space formula_15 is one of three components in a probabilistic model (a probability space). The other two basic elements are: a well-defined set of possible events (an event space), which is typically the power set of formula_15 if formula_15 is discrete or a σ-algebra on formula_15 if  it is continuous, and a probability assigned to each event (a probability measure function). A sample space can be represented visually by a rectangle, with the outcomes of the sample space denoted by points within the rectangle. The events may be represented by ovals, where the points enclosed within the oval make up the event. Conditions of a sample space. A set formula_19 with outcomes formula_20 (i.e. formula_21) must meet some conditions in order to be a sample space: For instance, in the trial of tossing a coin, one possible sample space is formula_28, where formula_5 is the outcome where  the coin lands heads and formula_6 is for tails. Another possible sample space could be formula_31. Here, formula_32 denotes a rainy day and formula_33 is a day where it is not raining. For most experiments, formula_34 would be a better choice than formula_35, as an experimenter likely do not care about how the weather affects the coin toss. Multiple sample spaces. For many experiments, there may be more than one plausible sample space available, depending on what result is of interest to the experimenter. For example, when drawing a card from a standard deck of fifty-two playing cards, one  possibility for the sample space could be the various ranks (Ace through King), while another could be the suits (clubs, diamonds, hearts, or spades). A more complete description of outcomes, however, could specify both the denomination and the suit, and a sample space describing each individual card can be constructed as the Cartesian product of the two sample spaces noted above (this space would contain fifty-two equally likely outcomes). Still other sample spaces are possible, such as right-side up or upside down, if some cards have been flipped when shuffling. Equally likely outcomes. Some treatments of probability assume  that the various outcomes of an experiment are always defined so as to be equally likely. For any sample space with formula_36 equally likely outcomes, each outcome is assigned the probability formula_37. However, there are experiments that are not easily described by a sample space of equally likely outcomes—for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no physical symmetry to suggest that the two outcomes should be equally likely. Though most random phenomena do not have equally likely outcomes, it can be  helpful to define a sample space in such a way that outcomes are at least approximately equally likely, since this condition significantly simplifies the computation of probabilities for events within the sample space. If each individual outcome occurs with the same probability, then the probability of any event becomes simply: For example, if two fair six-sided dice are thrown to generate two uniformly distributed integers, formula_39 and formula_40, each in the range from 1 to 6, inclusive, the 36 possible ordered pairs of outcomes formula_41 constitute a sample space of equally likely events. In this case, the above formula  applies, such as calculating the probability of a particular sum of the two rolls in an outcome. The probability of the event that the sum formula_42 is five is formula_43, since four of the thirty-six equally likely pairs of outcomes sum to five. If the sample space was the all of the possible sums obtained from rolling two six-sided dice, the above formula can still be applied because the dice rolls are fair, but the number of outcomes in a given event will vary. A sum of two can occur with the outcome formula_44, so the probability is  formula_45. For a sum of seven, the outcomes in the event are formula_46, so the probability is formula_47. Simple random sample. In statistics, inferences are made about characteristics of a population by studying a sample of that population's individuals. In order to arrive at a sample that presents an unbiased estimate of the true characteristics of the population, statisticians often seek to study a simple random sample—that is, a sample in which every individual in the population is equally likely to be included. The result of this is that every possible combination of individuals who could be chosen  for the sample has an equal chance to be the sample that is selected (that is, the space of simple random samples of a given size from a given population is composed of equally likely outcomes). Infinitely large sample spaces. In an elementary approach to probability, any subset of the sample space is usually called an event. However, this gives rise to problems when the sample space is continuous, so that a more precise definition of an event is necessary. Under this definition only measurable subsets of the sample space, constituting a σ-algebra over the sample space itself, are  considered events. An example of an infinitely large sample space is measuring the lifetime of a light bulb. The corresponding sample space would be .