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Given two events A and B from the sigma-field of a probability space, with the unconditional probability of B being greater than zero (i.e., P(B) > 0), the conditional probability of A given B (()) is the probability of A occurring if B has or is assumed to have happened. [5]
Then the unconditional probability that = is 3/6 = 1/2 (since there are six possible rolls of the dice, of which three are even), whereas the probability that = conditional on = is 1/3 (since there are three possible prime number rolls—2, 3, and 5—of which one is even).
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to ...
Since the probability of given is the same as the probability of given both and , this equality expresses that contributes nothing to the certainty of . In this case, A {\displaystyle A} and B {\displaystyle B} are said to be conditionally independent given C {\displaystyle C} , written symbolically as: ( A ⊥ ⊥ B ∣ C ) {\displaystyle (A ...
Prisoner A, prior to hearing from the warden, estimates his chances of being pardoned as 1 / 3 , the same as both B and C. As the warden says B will be executed, it is either because C will be pardoned ( 1 / 3 chance), or A will be pardoned ( 1 / 3 chance) and the coin to decide whether to name B or C the warden flipped ...
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".
Conditional dependence of A and B given C is the logical negation of conditional independence (()). [6] In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event. [7]
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.