Query: Your odds of correctly guessing the outcome of a flipped coin are 1 in 2, but your odds of guessing correctly twice in a row are only 1 in 4 —

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Probability <=SEP=> Probability is the measure of the likelihood that an event will occur . Probability is quantified as a number between 0 and 1 , where , loosely speaking , 0 indicates impossibility and 1 indicates certainty . The higher the probability of an event , the more certain that the event will occur . A simple example is the tossing of a fair ( unbiased ) coin . Since the coin is fair , the two outcomes ( `` heads '' and `` tails '' ) are both equally probable ; the probability of `` heads '' equals the probability of `` tails '' ; and since no other outcomes are possible , the probability of either `` heads '' or `` tails '' is 1/2 ( which could also be written as 0.5 or 50 % ) .   These concepts have been given an axiomatic mathematical formalization in probability theory , which is used widely in such areas of study as mathematics , statistics , finance , gambling , science ( in particular physics ) , artificial intelligence/machine learning , computer science , game theory , and philosophy to , for example , draw inferences about the expected frequency of events . Probability theory is also used to describe the underlying mechanics and regularities of complex systems .

Gambler's fallacy <=SEP=> The gambler 's fallacy , also known as the Monte Carlo fallacy or the fallacy of the maturity of chances , is the mistaken belief that , if something happens more frequently than normal during some period , it will happen less frequently in the future , or that , if something happens less frequently than normal during some period , it will happen more frequently in the future ( presumably as a means of balancing nature ) . In situations where what is being observed is truly random ( i.e. , independent trials of a random process ) , this belief , though appealing to the human mind , is false . This fallacy can arise in many practical situations , but is most strongly associated with gambling , where such mistakes are common among players .   The use of the term Monte Carlo fallacy originates from the most famous example of this phenomenon , which occurred in a Monte Carlo Casino in 1913 .

Random variable <=SEP=> In probability and statistics , a random variable , random quantity , aleatory variable , or stochastic variable is a variable quantity whose value depends on possible outcomes . As a function , a random variable is required to be measurable , which rules out certain pathological cases where the quantity which the random variable returns is infinitely sensitive to small changes in the outcome .   It is common that these outcomes depend on some physical variables that are not well understood . For example ,  when you toss a coin , the final outcome of heads or tails depends on the uncertain physics . Which outcome will be observed is not certain . Of course  the coin could get caught in a crack in the floor , but such a possibility is excluded from consideration . The domain of a random variable is the set of possible outcomes . In the case of the coin , there are only two possible outcomes , namely heads or tails . Since one of these outcomes must occur , either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability .   A random variable is defined as a function that maps outcomes to numerical quantities ( labels ) , typically real numbers . In this sense , it is a procedure for assigning a numerical quantity to each physical outcome , and , contrary to its name , this procedure itself is neither random nor variable . What is random is the unstable physics that describes how the coin lands , and the uncertainty of which outcome will actually be observed .   A random variable 's possible values might represent the possible outcomes of a yet-to-be-performed experiment , or the possible outcomes of a past experiment whose already-existing value is uncertain ( for example , due to imprecise measurements or quantum uncertainty ) . They may also conceptually represent either the results of an `` objectively '' random process ( such as rolling a die ) or the `` subjective '' randomness that results from incomplete knowledge of a quantity . The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself but is instead related to philosophical arguments over the interpretation of probability . The mathematics works the same regardless of the particular interpretation in use .   A random variable has a probability distribution , which specifies the probability that its value falls in any given interval . Random variables can be discrete , that is , taking any of a specified finite or countable list of values , endowed with a probability mass function characteristic of the random variable 's probability distribution ; or continuous , taking any numerical value in an interval or collection of intervals , via a probability density function that is characteristic of the random variable 's probability distribution ; or a mixture of both types . Two random variables with the same probability distribution can still differ in terms of their associations with , or independence from , other random variables . The realizations of a random variable , that is , the results of randomly choosing values according to the variable 's probability distribution function , are called random variates .   The formal mathematical treatment of random variables is a topic in probability theory . In that context , a random variable is understood as a function defined on a sample space whose outputs are numerical values .