Trial By ProbabilityBayes’ Theorem in Court

Presented By...

• Dave Bucheger

• Jill Thompson

• Sally Danielson

• Justin Koplitz

• Eric Hartmann

Presentation Overview

• 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?

• 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).

To find out, studies involving hypothetical crimes and mock juries were conducted:

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

Phosphate

A T

C G

Deoxyribose

• Has 3 billion places that tell traits

• Much code identical

• Unique places are called DNA markers

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

• = probability of innocence given all the evidence

• Upper bound can be found

Compounding Evidence

• 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 AnalogyObtained from Crime Scene

Defendant’s Blood Sample

Conclusion

• 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