Trial By Probability
Bayes’ Theorem in Court
Presented By...
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Dave Bucheger
Jill Thompson
Sally Danielson
Justin Koplitz
Eric Hartmann
Presentation Overview
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Background of current DNA cases
Do jury members understand probabilities?
What actually is DNA?
Conditional Probability/Compounding
Evidence
• O.J. Simpson Case
• DNA analogy
DNA In The News
• First case DNA conviction 1979.
• DNA strands on dog hair match strands on
duffel bag
• Serial killer linked to past murders
• Australian police testing 600 men over the
age of 18
• British propose DNA database
Illinois Cases
• Governor suspended death
penalty
• 13 inmates cleared
• 12 put to death
• 64 criminal convictions
cleared nationally from
DNA testing
More DNA & Probabilistic Evidence in Court
Andrew Dean Retrial
• Dean convicted of rape in 1990 with help of DNA
match probabilities.
• 1994 retrial questions presentation of the DNA
probabilities.
• Forensic evidence = P( Match | Innocent).
• Jury’s question = P( Innocent | Match).
• P(Match | Innocent) = P(Innocent | Match) ==>
PROSECUTOR’S FALLACY!
Prosecutor’s Fallacy Example:
• Assume there’s a poker game between several judges
and the Archbishop of Canterbury.
• P(Archbishop dealing royal flush | honest) = 1/70,000.
• P(Archbishop honest | dealt royal flush) > 1/70,000.
• Prosecutor’s fallacy equates the two probabilities, but
P(dealing a flush | honest ) is not necessarily equal to
P( honest | dealing a flush ).
Using Bayes’ Theorem for Probabilistic
Evidence
• We want to find the “odds” of the defendant being
innocent.
• Ratio of the probability of the innocence to the
probability of guilt.
• “Prior odds” = odds of innocence before jury hears
DNA evidence.
• “Posterior odds” = odds of innocence after jury hears
DNA evidence.
• Posterior odds = prior odds x DNA match probability.
Match Probability Confusion:
• Pretend a crime is committed in Menomonie by a white
male.
• Prior odds = 8,000 to 1 in favor innocence.
• Probability of DNA match from crime scene & suspect is
1 in a million.
• Posterior odds = 8,000 x 1 in a million, or 125 to 1 in
favor of innocence.
• Posterior odds are not 1 in a million!
Results of Dean Case:
• Idea of “Prosecutor’s Fallacy” is effective
enough to lift conviction and order a retrial.
• Idea that DNA evidence should be presented
carefully in court to avoid the Prosecutor’s
Fallacy.
Do Jury Members Understand
Probabilistic Evidence?
To find out, studies involving hypothetical
crimes and mock juries were conducted:
• Juries initially write down their probability of how
guilty they think the defendant is (labeled P0).
• Probabilistic evidence is presented to the jury like
frequency of an incriminating trait occurring in the
random population (or F(T)), or the probability of
a suspect having the trait (or P(TD)).
• Posterior probabilities are taken from the jury after
the evidence is presented (labeled P1).
Data from the juries is compared with Bayes’
Theorem probabilities of guilt:
Odds of guilt or P (G):
We can substitute 1/F (T) for the likelihood ratio, so the
probability of guilt is now:
Comparing Actual Jury Statistics to the Bayes’
Probabilities
“Trial”
F (T)
Mean P0
Mean P1
Bayes P1
Goodman
.001
.29
.47
.997
Goodman
.10
.29
.34
.803
Faigmon,
Baglioni
.20
.61
.70
.876
Faigmon,
Baglioni
.40
.64
.71
.816
Results of Chart
• The mean P1 and the Bayes P1 show the jury
underweighs probabilistic evidence.
• Prosecutor’s Fallacy of P( TD | G ) = P ( G | TD )
does not arise.
• Here’s a case where probabilistic evidence was
severely underweighed:
O.J. Simpson Trial
• Double murder case
• Blood samples found
– In driveway, foyer, bathroom
– Simpson’s Ford Bronco, sock, gloves
• Defense’s final claim - evidence planted
• Simpson found not guilty
What is DNA?
• Double Helix Form
• Has 3 billion places
that tell traits
• Much code identical
• Unique places are
called DNA markers
A
T
Phosphate
C
G
Deoxyribos
e
DNA Fingerprinting
• Compares DNA samples
• DNA separated by
– detergent
– pressure
• Sample gets segmented
• Rendered to a single
strand
Conditional Probability
• M = DNA match of
defendants blood and blood
at crime scene
• I = event defendant is
innocent
• I’ = event defendant is guilty
Conditional Probability
• P(M | I) = match of two individual’s blood
sample, very low chance, 10-8 or 10-10
• P(I | M) = probability of innocence given the
evidence
• Bayes’ theorem gives
• So P(I|M)
P(M|I) unless P(I) = P(M)
Conditional Probability
Compounding Evidence
• M1, M2, …, Mk = all evidence introduced
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= probability of innocence
given all the evidence
• Upper bound can be found
Compounding Evidence
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for a given event I
• “Conditionally Independent” means
• “Strongly Associated” measured by the
ratio
Compounding Evidence
• If < 1, M2 less likely given M1
• If > 1, M2 more likely given M1
• If = 1, M2 and M1 are mutually independent
Application To Simpson Case
• M1 = blood near victim is
consistent with defendant
• M2 = blood found on
defendant’s sock is
consistent with victim’s
• NOTE: (compliment of world’s population)
Application To Simpson Case
...Translation
• Probability of innocence
given the defendant’s blood
matched in the two separate
events
9 people in a billion!
Metaphorical Analogy
• 52!
possible card permutations
• DNA permutations much larger
• Not all information can be determined
depending on degree
• Analogous to a card if only number, suit, or
color could be determined
Metaphorical Analogy
Obtained from Crime Scene
Defendant’s Blood Sample
Conclusion
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DNA involved in many court cases
Possible DNA permutations are monstrous
Evidence not presented correctly
O.J. guiltier than sin
Any questions?
Works Cited
• Dale, Mike. “Evidence: The True Witness.”
[http://library.thinkquest.org/17049/]. 1998.
• Saunders, Sam C. “Compounding Evidence from Multiple
DNA-Tests.” Mathematics Magazine. 72, NO1. Feb. 1999:
39-43.
• Starr, Cecie. Biology Concepts and Applications. 4th Ed.
Brooks/Cole, 2000.
Special Thanks
• Steve Deckelman - Models Professor
• Steve Nold - Biology Professor, DNA Guru
• Eric Larson - Down right smart guy