Sample Space S = {a,b,...,z} The set of all possible outcomes of a statistical experiment, and each individual outcome is refered to as a element, member, or sample point
An Event is a subset of a sample space, and a compliment to event is the subset encompassing all that is not within the focus event.
The intersection of two events or (A ∩ B) is the event containing all elements that are common to A and B
Two events A and B are mutually exclusive or disjoint then A and B have no elements in common or (A ∩ B = φ)
The union of the two sample space subsets, or events A and B, denoted by (A∪B) is the event containing all the elemntes that belong to A, B, or both
Generalized Multiplication Rule If an operation can be performed in n1 ways, and if for each of these a second operation can be performed in n2 ways, and for each of the first two a third operation can be performed in n3 ways, and so forth, then the sequence of k operations can be performed in n1n2 ··· nk ways.
A Permutation is an arrangement of all or part of a set of objects, and The number of permutations of "n factorial" objects is n! such that n! = n(n-1)...(2)(1)
Also, in general, n distinct objects taken r at a time can be arranged in n(n − 1)(n − 2)···(n − r + 1) ways and is represented as nPr = (n! / (n − r)!)
Circular Permutations are permutations that occur by arranging objects in a circle, and in number is (n - 1) !. With this in mind, The number of distinct permutations of n things of which n1 are of one kind, n2 of a second kind, ... , nk of a kth kind is (n! / n1!n2! ··· nk!)
When you make r partitioned subsets of n objects those subsets are called cells
The number of ways of selecting r objects from a set of n objects without regard to order is refered to as the number of Combinations. A combination is actually a partition with two cells, the one containing the r objects selected and the other containing the (n-r) left: (n r) = ( n! / r!(n − r) )
Weights or Probabilities
The likelihood of the occurrence of an event resulting from such a statistical experiment is evaluated by means of real numbers
Probability is the sum of all sample points in an event P(A) = P(A1) + P(A2) + ... + P(A1)
Or P(A) = ( n / N ) If an experiment can result in any one of N different equally likely outcomes, and if exactly n of these outcomes correspond to event A
Relative Frequency
How often something happens divided by all outcomes
Indifference
There are no external influences on the probability of any one outcome from a process
Subjective
use of intuition, personal beliefs, and other indirect information in arriving at probabilities
Additive Rule states that the probability of event A or event B is the sum of the probabilities of event A and the probabilities of event b excluding the probability of them both occurring at the same time i.e:
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
In the event that the two events are mutually exclusive:
P(A ∪ B) = P(A) + P(B)
Multiplication/Product Rule The probability that two events both occur:
P(A ∩ B) = P(A)P(B|A), provided P(A) > 0.
Conditional Probability
The conditional probability of B, given A, denoted by P(B|A), is defined by:
P(B|A) = ( P(A ∩ B) / P(A) ) , provided P(A) > 0
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