Probability Theory and Likelihood Ratios - strbase.nist.gov

1 NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute1 Probability Theory and Likelihood RatiosSimone Gittelson, , Coble, , DNA Mixture Interpretation WorkshopAllentown, PASeptember 25-26, 2015 AcknowledgementI thankMichael Coble, Bruce Weir and John Buckleton for of viewin thispresentationare mine and do not necessarilyrepresentthe official position or policiesof the National Institute of Standards and Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute2 Probability TheoryVocabulary and Notationrolling a 6 and a 1it will snow tomorrowobserving a peak for allele 8Mr.

2 X stabbed Mr. Y8 NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute3 EVENTPROPOSITIONV ocabulary and Notation 8 COMPLEMENT event orpropositionVocabulary and Notation But I do not know which one is true?!? I am Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute4 PROBABILITYV ocabulary and NotationPr( )Pr( )Pr( )Pr( ) of ProbabilitycertainLaw #1:A Probability can take any value between 0 and 1, including 0 and 1. EXAMPLE: rollinga 6-sided die 1,2,3,4,5 6=1 7=0 NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute5 Laws of ProbabilityLaw #2: The Probability of event A orevent B occurring is equal to the Probability of event A plusthe Probability of event Bminus the Probability of event A and Aevent BPr =Pr +Pr Pr( ) = + A: rolling an odd number B: rolling a 2 = + =1 Second Law of is the Probability that a person drawn randomly from this population is blood group type A or AB?

3 Is the Probability that a person drawn randomly from this population is not blood group type AB?Blood groups:data from: Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training is the Probability that a person drawn randomly from this population is blood group type A or AB? groups:2. What is the Probability that a person drawn randomly from this population is not blood group type AB? groups:NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute7event Aevent BLaws of ProbabilityLaw #2: The Probability of event A orevent B occurring is equal to the Probability of event A plusthe Probability of event Bminus the Probability of event A and B.

4 = + A: rolling an odd number B: rolling a number greater than 2Pr =Pr +Pr Pr( )Second Law of is the Probability that a person drawn randomly from this population has a blood group that contains an A (groups A and AB) or a blood group that contains a B (groups B and AB)?Blood groups:data from: Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training is the Probability that a person drawn randomly from this population has a blood group that contains an A (groups A and AB) or a blood group that contains a B (groups B and AB)? groups:event Aevent BLaws of ProbabilityLaw #3: The Probability of event A andevent B occurring is equal to the Probability of event A timesthe Probability of event =Pr Pr | =Pr( ) Pr( | )A: rolling an odd number B: rolling a 3 or lessNEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute9 How Tall is Stefanie?

5 Conditional ProbabilitiesWhat is ?What is | ?AB1001040101001040 NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute10 Third Law of Probability (independent events) + is the Probability that a person drawn randomly from this population is blood group type A and Rh+?Blood groups:Rh factor:data from: is the Probability that a person drawn randomly from this population is blood group type A and Rh+? + groups:Rh factor:NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute11 Third Law of is the Probability that a person drawn randomly from this population has blond hair and blue eyes?

6 Color:eye color:data from: Law of Probability (dependent events) is the Probability that a person drawn randomly from this population has blond hair and blue eyes? color of blond haired color:data from: Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training is the Probability that a person drawn randomly from this population has blond hair and blue eyes? color of blond haired color:Hardy-Weinberg LawHomozygotefathermother28{28,28}28= =28 =28= 28 28= 282 28,28 28= 28,28= Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute13 Hardy-Weinberg LawHeterozygotefathermother13{13,16}16= =13 =16+ =16 =13= 13 16+ 16 13= 13 16+ 16 13=2 13 16 13,16or 16or 13 Hardy-Weinberg LawHeterozygotefathermother{13,16} 13= 16= 13.

7 16= 16or 13 NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute14 Law of total probabilityor the extension of the conversation:The Probability of event A can be partitioned into the sum of probabilities of event A conditioned on mutually exclusive and exhaustive of Probabilityevent Aevent BPr( )=Pr Pr +Pr( | ) Pr( )Law of Total is the Probability that a person drawn randomly from this population has blue eyes? color:NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training is the Probability that a person drawn randomly from this population has blue eyes?

8 : the animal isan elephant : the animal has four legsPr( | )4 legs1 leg2 legs3 legsgiven:~1 Conditional Probability ?Pr( | )4 legs15,000given:NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute16 : This man is Santa Claus. :This man has a white ( | )=? Probability that this man is Santa Claus given that he has a white beard. Probability that this man has a white beard given that he is Santa A) and B) are of the am completely Probability that this man is Santa Claus given that he has a white beard. Probability that this man has a white beard given that he is Santa A) and B) are of the am completely lost : This man is Santa Claus.

9 :This man has a white ( | )=? Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training am completely : This man is Santa Claus. :This man has a white Pr( | )= Pr( | )? Likelihood RatioNEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute18 Single ContributorStaincrime scenepersonof interest(POI) Single ContributorStaincrime scenepersonof interest(POI) 88 NEAFS Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute19 There are twosidesto crime staincame fromthe personof interest(POI).

10 The crime staindidnot come fromthe POI. It came spropositiondefense sproposition : :Conditional ProbabilityPr( | ) Pr( | )givengiven88 POIcrimestain88 POIcrimestain : The crime staincame fromthe POI. : The DNA typingresultsof the crime stainand the POI ssamplebothshow a peakfor Probabilistic DNA Mixture Interpretation Workshop9/25/2015 Cedar Crest CollegeForensic Science Training Institute20We have the DNA typing results of only one marker. This marker is called NEW, and we don t know anything about the alleles at this scene:Albert:8 NEW: 8 Does the DNA on the crime scene come from Albert?A)YesB)NoC)I need more informationD)I am completely lostWe have the DNA typing results of only one marker.