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In probability theory, an outcome is a possible result of an experiment or trial. [1] 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. [2]
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%).
The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value .
As a discrete probability space, the probability of any particular lottery outcome is atomic, meaning it is greater than zero. Therefore, the probability of any event is the sum of probabilities of the outcomes of the event. This makes it easy to calculate quantities of interest from information theory.
For any sample space with equally likely outcomes, each outcome is assigned the probability . [16] 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 ...
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.
The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables. [ citation needed ] One author uses the terminology of the "Rule of Average Conditional Probabilities", [ 4 ] while another refers to it as the "continuous law of ...
Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p.Three examples are shown: Blue curve: Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to ...