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The events 1 and 6 are mutually exclusive but not collectively exhaustive. The events "even" (2,4 or 6) and "not-6" (1,2,3,4, or 5) are also collectively exhaustive but not mutually exclusive. In some forms of mutual exclusion only one event can ever occur, whether collectively exhaustive or not.
The event calculus is a logical theory for representing and reasoning about events and about the way in which they change the state of some real or artificial world. It deals both with action events, which are performed by agents, and with external events, which are outside the control of any agent.
In probability theory, an event is a subset of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. [1] A single outcome may be an element of many different events, [2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. [3]
The probabilities of the individual events (red, and club) are multiplied rather than added. The probability of drawing a red and a club in two drawings without replacement is then 26/52 × 13/51 × 2 = 676/2652, or 13/51. With replacement, the probability would be 26/52 × 13/52 × 2 = 676/2704, or 13/52.
From the Kolmogorov axioms, one can deduce other useful rules for studying probabilities. The proofs [6] [7] [8] of these rules are a very insightful procedure that illustrates the power of the third axiom, and its interaction with the prior two axioms. Four of the immediate corollaries and their proofs are shown below:
In probability theory, a conditional event algebra (CEA) is an alternative to a standard, Boolean algebra of possible events (a set of possible events related to one another by the familiar operations and, or, and not) that contains not just ordinary events but also conditional events that have the form "if A, then B".
There are many and varied events, objects, and facts which require explanation. So too, there are many different things that can be used to explain something. Aristotle recognized four archetypes of explanation. These were thought, since even more ancient times, to be universal and unique 'kinds' of explanation that comprise all ways of ...
The transition probabilities depend only on the current position, not on the manner in which the position was reached. For example, the transition probabilities from 5 to 4 and 5 to 6 are both 0.5, and all other transition probabilities from 5 are 0. These probabilities are independent of whether the system was previously in 4 or 6.