Probab and Statistics I
Probab and Statistics I MATH 4339
University of Texas-Pan American (UTPA)
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Date Created: 10/29/15
Chap 2 Axioms of Probability Motivating Example EX A 5card poker hand is said to be a full house if it consists of 3 cards of the same denomination and 2 cards of the same denomination That is a full house is three of a kind plus a pair What is the probability that one is dealt a full house 22 Sample Space and Events A pair of concepts Sample space the set of all possible outcomes of an experiment Denoted by S Event any subset of the sample space is know as an event Denoted by E EX First row a die then toss a coin What is the sample space Example on textbook bkpg 2223 Union Intersection and Complement 1 union E U F 2 Intersection E I F 3 Complement If Venn diagram Example The following table summarizes visitors to a local amusement park AllDay HalfDay P Pass ass Total Male 1200 800 2000 Female 900 700 1600 Total 2100 1500 3600 One visitor from this group is selected at random 1 De ne the event A as the visitor purchased an allday pass 2 De ne the event B as the visitor selected purchased a halfday pass 3 De ne the event C as the visitor selected is female Describe the relationship and draw Venn diagram Rules for operation of forming Unions Intersections and Complements 1 Commutative lawsEUFFUE E FF E 2 Associative laws E UF U GEU FU G E F GE Ff G 3 DistributivelawsEUF GE GUF G 4 DeMorgan s laws quotWEIC OE i1 i1 i1 i1 We will come back to these rules when we need them later in this chapter 23 Axioms of Probability n E Relative frequency de nition of probability P E 11mg quot900 n Here an experiment with sample space S is repeatedly performed under same conditions E is an event in the sample space nE is the number of times in first 71 repetitions of the experiment that the event E occurs Axiom l3 bkpg 27 Examples on textbook bkpg 28 24 Some Simple Propositions PEC l P E IfEC F thenPE SPF PE U F PE PE PE W F General Addition Rule If E and F are mutually exclusive then PE U F PE PF Special Addition Rule ie DeMorgan s laws Example A consumer is selected at random The probability the consumer has tried a snack food F is 05 tried a new soft drink D is 06 and tried both the snack food and the soft drink is 02 Find the following probabilities l Pdid not tried snack food 2 Ptried neither snack food nor soft drink 3 P tried snack food but not soft drink 4 P tried snack food or soft drink 25 Sample Spaces Having Equally Likely Outcomes Now we consider the situation all outcomes in the sample space are equally likely to occur Say the sample space is S and E is the event in the sample space then number 0 outcomes in E PE f number of outcomes in S Examples on textbook bkpg 34 35 37 39 27 Probability as A Measure of Belief Read the textbook bkpg 48