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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.
A is assumed to be the set of all possible outcomes of an experiment or random trial that has a restricted or reduced sample space. The conditional probability can be found by the quotient of the probability of the joint intersection of events A and B, that is, (), the probability at which A and B occur together, and the probability of B: [2 ...
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]
For example, the probability of the union of the mutually exclusive events and in the random experiment of one coin toss, (), is the sum of probability for and the probability for , () + (). Second, the probability of the sample space Ω {\displaystyle \Omega } must be equal to 1 (which accounts for the fact that, given an execution of the ...
To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events that contain no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that any of these events occurs is given by the sum of the ...
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 ...
Similar to the examples described above, we consider x, y, φ to be independent uniform random variables over the ranges 0 ≤ x ≤ a, 0 ≤ y ≤ b, − π / 2 ≤ φ ≤ π / 2 . To solve such a problem, we first compute the probability that the needle crosses no lines, and then we take its complement.
The probability that X n = 0 occurs for infinitely many n is equivalent to the probability of the intersection of infinitely many [X n = 0] events. The intersection of infinitely many such events is a set of outcomes common to all of them. However, the sum ΣPr(X n = 0) converges to π 2 /6 ≈ 1.645 < ∞, and so the Borel–Cantelli Lemma ...