Math 4030-2a - Sample Space - Events - Definition of Probabilities Math 4030-2a - Sample Space - Events - Definition of Probabilities (of Events) 5/12/2018

Random Experiment An experiment is called Random experiment if The outcome of the experiment in not known in advance All possible outcomes of the experiment are known. 5/12/2018

Sample space and events Set of all possible outcomes of an experiment is called sample space We will denote a sample space by S finite or infinite. discrete or continuous. Any subset of a sample space is called an event. 5/12/2018

Operations on events Mutually Exclusive Events Union, , “or” Intersections,  , “and” Complement, , “not” Venn diagram 5/12/2018

How many outcomes in the sample space? Tree diagram: Identify the stages in the experiment; Identify possible outcomes at every stage; Count the number of “leaves” , for the size of the sample space. 5/12/2018

Multiplication Rule: k stages; there are n1 outcomes at the 1st stage; from each outcome at ith stage, there are ni outcomes at (i+1)st stage; i=1,2,…,k-1. Total number of outcomes at kth stage is 5/12/2018

Permutation Rule: n distinct objects; take r (<= n) to form an ordered sequence; Total number of different sequences is 5/12/2018

Factorial notation: Permutation number when n = r, i.e. 5/12/2018

Combination Rule: n distinctive object; take r (<= n) to form a GROUP (with no required order) Total number of different groups is 5/12/2018

Count without counting: Multiplication: Independency between stages; Permutation: Choose r from n (distinct letters) to make an ordered list (words). Special case of multiplication; Factorial: Special case of permutation; Combination: Choose r from n, with no order. 5/12/2018

Probability of an event P(A) Event A S [0,1] 5/12/2018

Axioms of probability: Axiom 1. 0 ≤ P(A) ≤ 1. Axiom 2. P(S) = 1 Axiom 3. If A and B are mutually exclusive events then P(A U B) = P(A) + P(B) 5/12/2018

Axioms of probability Generalization of Axiom 3. If A1, A2, …, An are mutually exclusive events in a sample space S then P(A1 U A2 U … U An) = P(A1) + P(A2) + … + P(An) 5/12/2018

Addition rule of probability If A and B are any events in S then P(AUB) = P(A) + P(B) – P(A  B) Special case: if A and B are mutually exclusive, then P(AUB) = P(A) + P(B). 5/12/2018

Classical probability has assumptions: There are m outcomes in a sample space (as the result of a random experiment); All outcomes are equally likely to occur; An event A (of our interest) consists of s outcomes; Then the definition of the probability for event A is 5/12/2018

Probability rule of the complement If B is the complement of A, then P(B) = 1 - P(A). 5/12/2018

Relative frequency approach Perform the experiment (trial) m times repeatedly; Record the number of experiments/trials that the desired event is observed, say s; Then the probability of the event A can be approximated by 5/12/2018