10/1/20151 Math a Sample Space, Events, and Probabilities of Events

10/1/20152 Random Experiment An experiment is called Random experiment if 1.The outcome of the experiment in not known in advance 2.All possible outcomes of the experiment are known.

10/1/20153 Sample space and events (Sec. 3.1) 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.

10/1/20154 Operations on events  Union,, “or”  Intersections,, “and”  Complement,, “not”  Mutually Exclusive Events  Venn diagram

Probability of an event (Sec. 3.3) 10/1/20155 Event A P(A) S

Axioms of probability (Sec. 3.4) 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) 10/1/20156

Axioms of probability Generalization of Axiom 3. If A 1, A 2, …, A n are mutually exclusive events in a sample space S then P(A 1 U A 2 U … U A n ) = P(A 1 ) + P(A 2 ) + … + P(A n ) 10/1/20157

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). 10/1/20158

Probability rule of the complement 10/1/20159 If B is the complement of A, then P(B) = 1 - P(A).

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 10/1/201510

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 10/1/201511

1210/1/2015 Count without counting: Sample Space Event Pr(Event) = Count!

1310/1/2015 Multiplication Rule (P50): k stages; there are n 1 outcomes at the 1 st stage; from each outcome at ith stage, there are n i outcomes at (i+1)st stage; i=1,2,…,k-1. Total number of outcomes at kth stage is

1410/1/2015 Permutation Rule (P51): n distinct objects; take r (<= n) to form an ordered sequence; Total number of different sequences is

1510/1/2015 Factorial notation: Permutation number when n = r, i.e.

1610/1/2015 Combination Rule (P52): n distinctive object; take r (<= n) to form a GROUP (with no required order) Total number of different groups is

Count without counting: 10/1/ 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.