GradeBuddy AEM 201 1ST EditionLecture 9PREVIOUS LECTUREI. Other Characteristics of Data Distribution ShapesII. Other Tools for Exploratory Data Analysis (EDA)III. Measures of Association Between Two VariablesIV. Putting it All TogetherCURRENT LECTUREI. Introduction To ProbabilityII. Counting RulesIII. Methods Of Assigning Probability To Experimental OutcomesIV. Relationship Of ProbabilityINTRODUCTION TO PROBABILITY- Experiment: a replicable activity or process that results in a well-defined outcome. Experimental outcomes are denoted as Ei..- Trial: a single execution of an experiment- Sample Point: the outcome of the trial- Sample Space: the set of all possible unique sample points. Often denoted S={}- Probability: a numerical measure of the relative likelihood that an event will occur. The probability of an outcome is denoted as P(Ei)- EXAMPLE: Flipping a coino Experiment=a single flip of a quartero Trial=flipping a quarter once and recording the outcomeo Sample space= {heads, tail} of S={h, t}o If we refer to the outcome ‘heads’ as E1 and ‘tails’ as E2, then we knowintuitively that P(E1)=0.5- Fair: all outcomes are equally likely - People make the biggest mistake with probabilities is assuming all results areequally likely. That is not always true- When you roll a die twice and add the outcomes, the 11 outcomes are not all equally likely to occur- The more we roll a die, the less likely extremes become and the more likely the middle values become- Tree Diagram: graphical tool used to depict the sample points for a compound experimentGradeBuddy COUNTING RULES- Counting Rules: efficient means by which we can calculate the exact number of potential outcomes for an experiment- Counting Rule for Multiple-Step Experiment: for k-step experiment with n1, n2…nk possible outcomes associated with the k-steps, the total number of potential outcomes is:o n1*n2*…*nk- Counting Rule for Combinations: when n objects are to be selected from a set of N objects and the order of selection DOES NOT MATTER (AB and BA are equivalent) the total number of potential outcomes is:- N!=Factorialo 10!=10*9*8*7*6*5*4*3*2*1- You can simplify these factorial problems a lot- Say we are randomly selecting two people from a group of ten finalists to do aresearch study. How many different possible outcomes?o 10!/(2!(10-2)!)o Simplified to: 10*9/2*1=45- Counting Rule for Permutations: when n objects are to be selected from a set of N objects and the order of selection DOES MATTER- There will be more outcomes for a permutation- Two basic requirements for assigning probabilitites:o Probabilities must be between 0 and 1 (inclusively) for event Ei we know that 0.00 less than or equal to P(Ei) is less than or equal to 1.00o The sum of the possibilities for all possible experimental outcomes MUST sum to 1. This happens because the experimental outcomes are mutuallyexclusive and collectively exhaustedGradeBuddy METHODS OF ASSIGNING PROBABILITIES TOEXPERIMENTAL OUTCOMES- Classical (or Priori) Approach: assume all events are equally likely to assign probabilities to eventso P(A)= 1/total number of possible outcomeso Originally suggested by Galileo and university of Pisa when answering questions for his highness Cosimo II of Tuscany- Relative Frequency (or Empirical) Approach: assign probabilities to events in terms of the proportion of times that an event is observed over a large number of identical trialso P(A)= #of trials in which A is observed/ total # of trials executedo Suggested by Jacques Bernoulli and represents the fundamental philosophical shift in attitudes about probability (some outcomes are not equally likely)o This is an estimate and will be more reliable if it’s based on many trials. Phenomena if often referred to as the Law of large numbers-represents fundamental weakness-how do you exactly represent a trial?- Subjective Approach: define a probability in terms of an individual’s or group’s belief, intuition, or judgment about the frequency of times an event will occuro P(A)=# of trials which we believe will occur/total # of trials we are consideringo We do this dailyo Suggested by John Maynard Keynes and Harold Jefferies o Represents a philosophical shift in attitudes about probabilitieso This is also an estimate and will generally be more reliable if it’s basedon well-informed judgment. It’s shortcoming is this reliance and resulting inconsistency of the estimateRELATIONSHIP OF PROBABILITY- Event: a collection of sample points- Compliment: for some event A, the event consisting of all sample points that do not belong to an event A. often denoted as A with a bar over the top of Ac or A’.o It’s probability is P(A’)=1.00-P(A)o And P(A’)+P(A)= 1.00- Mutually Exclusive Event: two or more events that cannot occur simultaneously- Collectively Exhausted Events: events that include all possible outcomes of anexperiment- Set: a collection of objects- Cardinality: the number of unique elements that comprise a set |A|GradeBuddy - Null (or Empty) Set: a collection that contains no objects (has cardinality of