Outcome (probability)

For other uses, seeOutcome.

Inprobability theory,anoutcomeis a possible result of anexperimentor trial.Each possible outcome of a particular experiment is unique, and different outcomes aremutually exclusive(only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of asample space.

For the experiment where we flip a coin twice, the four possibleoutcomesthat make up oursample spaceare (H, T), (T, H), (T, T) and (H, H), where "H" represents a "heads", and "T" represents a "tails". Outcomes should not be confused withevents,which aresets(or informally, "groups" ) of outcomes. For comparison, we could define an event to occur when "at least one 'heads'" is flipped in the experiment - that is, when the outcome contains at least one 'heads'. This event would contain all outcomes in the sample space except the element (T, T).

Sets of outcomes: events

Main article:Event (probability theory)

Since individual outcomes may be of little practical interest, or because there may be prohibitively (even infinitely) many of them, outcomes are grouped intosetsof outcomes that satisfy some condition, which are called "events."The collection of all such events is asigma-algebra.

An event containing exactly one outcome is called anelementary event.The event that contains all possible outcomes of an experiment is itssample space.A single outcome can be a part of many different events.

Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of thepower setof the sample space are defined as events). However, this approach does not work well in cases where the sample space isuncountably infinite(most notably when the outcome must be somereal number). So, when defining aprobability spaceit is possible, and often necessary, to exclude certain subsets of the sample space from being events.

Probability of an outcome

Outcomes may occur with probabilities that are between zero and one (inclusively). In adiscreteprobability distribution whosesample spaceis finite, each outcome is assigned a particular probability. In contrast, in acontinuousdistribution, individual outcomes all have zero probability, and non-zero probabilities can only be assigned to ranges of outcomes.

Some "mixed" distributions contain both stretches of continuous outcomes and some discrete outcomes; the discrete outcomes in such distributions can be calledatomsand can have non-zero probabilities.

Under themeasure-theoreticdefinition of aprobability space,the probability of an outcome need not even be defined. In particular, the set of events on which probability is defined may be someσ-algebraon$S$and not necessarily the fullpower set.

Equally likely outcomes

Flipping a coin leads to twooutcomesthat are almost equally likely.

Up or down? Flipping a brass tack leads to twooutcomesthat are not equally likely.

In somesample spaces,it is reasonable to estimate or assume that all outcomes in the space are equally likely (that they occur with equalprobability). For example, when tossing an ordinary coin, one typically assumes that the outcomes "head" and "tail" are equally likely to occur. An implicit assumption that all outcomes are equally likely underpins mostrandomizationtools used in commongames of chance(e.g. rollingdice,shufflingcards,spinning tops or wheels, drawinglots,etc.). Of course, players in such games can try to cheat by subtly introducing systematic deviations from equal likelihood (for example, withmarked cards,loadedor shaved dice, and other methods).

Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely.However, there are experiments that are not easily described by a set of equally likely outcomes— for example, if one were to toss athumb tackmany times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely.

See also

• Event (probability theory)– In statistics and probability theory, set of outcomes to which a probability is assigned
• Sample space– Set of all possible outcomes or results of a statistical trial or experiment
• Probability distribution– Mathematical function for the probability a given outcome occurs in an experiment
• Probability space– Mathematical concept
• Realization (probability)– Observed value of a random variable

External links

• Media related toOutcome (probability)at Wikimedia Commons