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Topic 7: Probability Theory - Coggle Diagram
Topic 7: Probability Theory
Basics of Probability Theory
Probability is a numerical measure of the likelihood that a specific event will occur when the experiment is performed.
Basic Terms
Experiment
Experiment is a process whose outcome is subject to uncertainty.
Outcome
Outcome is the observation of the experiment.
Sample Space
Sample space of an experiment is the set of all possible outcomes and its symbol is S.
Event
An event is a subset of the sample space.
Relations from Set Theory
Intersection Event
The intersection of two events A and B, denoted by A∩B, is the event containing all elements that are common to A and B. is the event that both A and B occur.
Union Event
The union of the events A and B, denoted by A∪B, is the event containing all the elements that belong to A or B or both. A∪ B is the event that A or B occurs.
Complement Event
The complement of an event A with respect to S is the set of all elements of S that are not in A.
We denote the complement of A by the symbol A', 𝐴̅ is the event that A does not occur.
Mutually Exclusive Event
Two events A and B are mutually exclusive or disjoint if they have no common outcomes. In other words, A∩B =∅
Concepts of Probability
Classical Probability
Suppose S is a sample space in which all outcomes are assumed to be equally likely, and A is an event. Then the probability of A, denoted by P(A), is: P(A) = number of outcomes / total number of outcome of the experiment.
Empirical Probability (Frequentist Interpretation)
The probability of an event is the proportion of times the event would occur when the experiment is run a large number of times.
Subjective Probability (Bayesiant Interpretation)
The probability assigned to an event is based on subjective judgment, experience, information and belief.
Axioms and Properties of Probability
The probability of getting any event MUST be within 0 and 1. (0 ≤ P(A) ≤1)
The SUM of the probability of all outcomes for an experiment is always 1.(P(S) =1)
Mutually Exclusive Events: If A and B are mutually exclusive events in S, then P(A∩ B) = 0.
Union Events
i. P(A∪B) = P(A) + P(B) - P(A∪B)
ii. If A and B are mutually exclusive events, then P(A ∪ B) = P(A) + P(B)
iii. If there are more than 2 events are mutually exclusive, P(A+ B +C...) + P(A) + P(B) + P(C) +...
Complement Events
i. P(A') = 1- P(A)
ii. P(A∪B)‘ = 1 - P(A∪ B) = P(A' ∩ B')
Counting Techniques and Probability
Product Rule
Total outcomes for the experiment = m x n x k
Combination
The number of combinations for selecting r from n distinct elements = nCr
Note: Order of the selected objects or elements is NOT important.
Permutation
The no. of permutations of n distinct objects taken r at a time = nPr
Note: Order of the selected objects or elements is important.
Conditional Probability
It is the probability that an event will occur given
that another event has already occurred.
Conditional probability of A given B:P(A|B) = P(A ∩ B) / P(B)