Slide 1Subjective ProbabilityClassical Approach to ProbabilityEmpirical Approach to ProbabilityFoundations of Empirical ProbabilityMutually vs. Nonmutually Exclusive EventsCalculating ProbabilityCalculating ProbabilityThe addition rule and odds of dyingThe addition rule and odds of dyingMultiplication Rule for Independent Events: p(A) x p(B)Joint and Marginal ProbabilitiesCalculating ProbabilitiesFrequency Table ExampleFrequency Table ExampleFrequency Table ExampleCalculating ProbabilitiesCalculating ProbabilitiesConditional ProbabilitiesConditional Probabilitiesp(B|A)≠p(A|B)p(B|A)≠p(A|B)Slide 23The Binomial DistributionMean and Standard Deviation for a Binomial Distribution.The Binomial Distribution Continued.ExampleExample ContinuedQuiz time.Probability: The Foundation of Inferential StatisticsOctober 14, 2009Subjective Probability•Throughout the course I have used the word probability.–Yet I have not defined it. Instead I have relied on the assumption that you all have a sense of probability.–The book calls this sense of probability “subjective probability”.Classical Approach to Probability•The mathematical Definition of ProbabilityorEmpirical Approach to Probability•In classical approach, the parameters are known–The numbers of cards in a deck. (e.g. the probability of drawing a king from a full deck of cards)•In the empirical approach the parameters are not known.–Instead we use samples to calculate estimates of relative probabilities.Foundations of Empirical Probability•Discrete vs. Continuous Variables–Discrete- a variable that is represented in whole numbers. •Decimals don’t make sense–Continuous- a variable where intermediate or fractional values are valid.•Sample Space–All possible outcomes that can occur•In Mendelian genetics, dominant recessive allele Aa located at one locus may have the following sample space: AA, aa, Aa.•If Hardy-Weinberg Equilibrium is not violated the relative frequencies in the population should be:•f(AA)= 1, p=.25•f(Aa)= 2, p=.5•f(aa)=1, p=.25Mutually vs. Nonmutually Exclusive Events•Mutually Exclusive–Events in our sample that cannot occur together or overlap.•Nonmutually exclusive events–Events in our sample that can occur together.–A joint probability is the degree in which a set of events do occur together in a sample.Calculating Probability•Probability can be expressed –in a percentage or relative frequency in 100. –As a decimal. •p= .5 is the same as, 50% chance, is the same as saying 50 in one hundred.•The addition rule: p(A or B)–It is used when we want to calculate the probability of selecting an element that has one or more conditions.Calculating Probability•The addition rule: p(A or B) continued–p(A and B) is the joint probability.–When the events are mutually exclusive p(A and B) = 0The addition rule and odds of dying•The odds of dying p=1.0•Usually you die of one cause.•What are the odds of dying of either a plane accident or a bicycling accident.•Bicycling accident= 1 in 4919•Air/space accident= 1 in 5051The addition rule and odds of dying•p(A or B)= .0002 + .0002 – 0= .0004 –-0 because these are mutually exclusive causes of death. •.4% chance of dying in a bicycling accident or a air/space accident.Multiplication Rule for Independent Events: p(A) x p(B)•Used to determine the probability of two or more events occurring at the same time that are mutually exclusive. •Example: You want to know what the odds are that you will win the lottery. You have to match all five numbers. The choices range from 1:40.• The probability for choosing the first number is 1 in 40, the second number 1 in 39, third number 1 in 38…Joint and Marginal Probabilities•Joint and marginal probabilities refer to the proportion of an event as a fraction of the total. •To calculate these probabilities we divide the frequency of the joint or marginal probability of two or more events by the total frequency.A Not A Marginal Prob.B p(A and B) p(A and B) p(B)Not B p(A and not B) p(not A and not B) p(not B)Marginal Prob. p(A) p(not A)Calculating ProbabilitiesA Not A Marginal Prob.B p=f(A and B)/tot p=f(A and B)/tot p(B)= f(B)/Total Not B p=f(A and not B)/tot p=f(not A and not B)/tot p(not B)= f(not B)/Total Marginal Prob.p(A)= f(A)/Total p(not A)= f(not A)/Total total/total= 1.00•The previous graph just gave you where the different types of probabilities are located on the chart. •This chart gives you the way you would calculate these probabilites.•You will be given a frequency for each cell (e.g. B= 20, not B=80, A= 48, not A= 52)•With this information you should be able to create a similar chart.Frequency Table ExampleA Not AB 20 48not B 80 52Frequency Table ExampleA Not A Marginal ProbabilityB 20 48 = 20 +48not B 80 52 = 80 + 52Marginal Probability =20 + 80 =48 +52Frequency Table ExampleA Not A Marginal ProbabilityB 20 48 68not B 80 52 132Marginal Probability 100 100 Total = 200Calculating ProbabilitiesA Not A Marginal Prob.B p=20/200 p=48/200 p(B)= 68/200Not B p=80/200 p=52/200 p(not B)=132/200Marginal Prob.100/200 100/200 200/200= 1.00Calculating ProbabilitiesA Not A Marginal Prob.B p(A and B)=.1 p(not A and B)=.24 p(B)= .34Not B p(A and not B)=.4 p(not A and not B)=.26 p(not B)=.66Marginal Prob.p(A)=.5 p(A).5 200/200= 1.00Conditional ProbabilitiesA Not AB(not B)•Conditional probabilities are used when categories are not mutually exclusive.•The “|” symbol means given. •Therefore the first cell p(A|B) reads the probability of picking A given B•example from book A= Alcohol Abuse B= drug abuse. p(A|B) means the probability of picking an alcohol abuser among drug abusers.Conditional ProbabilitiesA Not AB(not B)p(B|A)≠p(A|B)A Not A Marginal Prob.B p(A |B)=.1/.34 p(not A |B)=.24/.34 p(B)= .34Not B p(A |not B)=.4/.66 p(not A |not B)=.26/.66 p(not B)=.66Marginal Prob.p(B)=.5 p(A).5 200/200= 1.00A Not A Marginal Prob.B p(B |A)=.1/.5 p(B |not A)=.24/.5 p(B)= .34Not B p(not B |A)=.4/.5 p(not B |not A)=.26/.5 p(not B)=.66Marginal Prob. p(B)=.5 p(A).5 200/200= 1.00p(B|A)≠p(A|B)A Not A Marginal Prob.B p(A |B)=.294 p(not A |B)=.706 p(B)= .34Not B p(A |not B)=.606 p(not A |not B)=.394 p(not B)=.66Marginal Prob.p(B)=.5 p(A).5 200/200= 1.00A Not A Marginal Prob.B p(B |A)=.2 p(B |not A)=.48 p(B)= .34Not B p(not B |A)=.8 p(not B |not A)=.52 p(not B)=.66Marginal Prob. p(B)=.5 p(A).5 200/200= 1.00Determining Joint Probabilites when