Processing physical evidenceProcessing physical evidence

• discovering, recognizing and examining it;

• collecting, recording and identifying it;

• packaging, conveying and storing it;

• exhibiting it in court;

• disposing of it when the case is closed.

Lecture: Forensic Evidence and Lecture: Forensic Evidence and ProbabilityProbability

Characteristics of evidenceCharacteristics of evidence

• Class characteristics

• Individual characteristics

features that place the

item into a specific

category

features that distinguish one item from another of the same type

The standard deviation (SD) quantifies variability. If the data follow a bell-shaped Gaussian distribution, then 68% of the values lie within one SD of the mean (on either side) and 95% of the values lie within two SD of the mean. The SD is expressed in the same units as your data.

The arithmetic mean is the "standard" average, often simply called the "mean"

1% of women at age forty who participate in routine screening have breast cancer.  80% of women with breast cancer will get positive mammographies.  9.6% of women without breast cancer will also get positive mammographies.  A woman in this age group had a positive mammography in a routine screening.  What is the probability that she actually has breast cancer?

1% of women at age forty who participate in routine screening have breast cancer.  80% of women with breast cancer will get positive mammographies.  9.6% of women without breast cancer will also get positive mammographies.  A woman in this age group had a positive mammography in a routine screening.  What is the probability that she actually has breast cancer?

STATISTICAL SOLUTIONTo put it another way, before the mammography screening, the 10,000 women can be divided into two groups:•Group 1:  100 women with breast cancer.•Group 2:  9,900 women without breast cancer.

  After the mammography, one gets:*  80 women with breast cancer, and a positive mammography. i.e. 80% of 100* 950 women without  breast cancer, and a positive mammography. i.e. 9.6% of 9900

The probability that a patient with a positive mammogram has breast cancer is: # (breast cancer + positive mammorgraphy) / #(positive mammorgraphy ) = 80/(80+950) = 7.8%

1% of women at age forty who participate in routine screening have breast cancer.  80% of women with breast cancer will get positive mammographies.  9.6% of women without breast cancer will also get positive mammographies.  A woman in this age group had a positive mammography in a routine screening.  What is the probability that she actually has breast cancer?

BAYESIAN SOLUTION

The original proportion of patients with breast cancer is known as the prior probability:  P(C) = 1% and P(~C) = 99%The chance of a patient having a positive mammography given that she has cancer, and the chance that of a patient having a positive mammography given that she does not have cancer, are known as the two conditional probabilities.  Collectively information is often termed the liklehood ratio: P(M | C) = 80% i.e probability of +ve mammogram given that she has cancerP(M | ~C) = 9.6% i.e probability of +ve mammogram given that she does not

have cancerThe final answer - the estimated probability that a patient has breast cancer given that we know she has a positive result on her mammography - is known as the revised probability or the posterior probability.

1% of women at age forty who participate in routine screening have breast cancer.  80% of women with breast cancer will get positive mammographies.  9.6% of women without breast cancer will also get positive mammographies.  A woman in this age group had a positive mammography in a routine screening.  What is the probability that she actually has breast cancer?

prior probability x conditional probability = posterior probability

P(C) . P(M | C) = P(C | M)P(~C) P(M | ~C) P(~C | M)

0.01 . 0.8 = 0.008 = 800.99 0.096 0.095 950

the estimated odds that a patient has breast cancer given that we know she has a positive result on her mammography are 80 to 950

the estimated probability that a patient has breast cancer given that we know she has a positive result on her mammography is 80 / (80+950) = 7.8%

prior probability P(C) . P(~C)

The probability that the suspect is or is not guilty prior to presenting this evidence

conditional probability P(M | C) P(M | ~C)

Also called the Likelihood Ratio (LR) and represents the probability that this evidence would be present if the suspect is or is not guilty

posterior probability P(C | M) P(~C | M)

The probability that the suspect is or is not guilty given the evidence presented

Bayesian Probability• Problem#1

A suspect is seen fleeing the crime. The suspect is positively identified as being at least six feet tall and was wearing a nurse’s uniform. Exactly 5% of the male population is at least 6 feet tall, while 0.5% of the female population is at least 6 feet tall, and 98% of all nurses are female. What are the odds that the suspect is a male.

• Problem#21 million people in America have HIV/AIDS. HIV tests correctly identify a HIV infected person with a positive result 97.7% of the time. HIV tests correctly identify a non-HIV infected person with a negative result 92.6% of the time. If an American gets a positive HIV test result what are the odds that they are infected with HIV? (Assume an american population of 260 million)

• Problem#3Suppose that a barrel contains many small plastic eggs.  Some eggs are painted red and some are painted blue.  40% of the eggs in the bin contain pearls, and 60% contain nothing.   30% of eggs containing pearls are painted blue, and 10% of eggs containing nothing are painted blue.  What is the probability that a blue egg contains a pearl? 

• Problem#4There are 100 people in a room, 20 women and 80 men. 80% of women are blonde, while 30% of the men are blonde. The suspect has blonde hair and is definitely one of the people in the room. What are the odds that the suspect is a female.

• Problem#5The investigator on the case informs you that the odds that the suspect committed the crime are 2 to 1. Your DNA fingerprint analysis of the suspect’s blood gives a 1 in a million probability that it is a random match to the blood found at the crime scene. You also know that your lab has a 1 in a 1000 chance of a false positive. What are the odds that the blood found at the crime scene came from your suspect?

Defender’s Fallacy : P(S | M) = P(M | ~S) x sample population

Prosecutor’s Fallacy :P(S | M) = 1 - P(M | ~S)

A crime has been committed, and a blood sample has been found at the crime scene. The blood is typed as A- , a blood type found in 5% of the population A suspect is identified, who also happens to have the A- blood type. In addition a DNA profile of the suspect gives the odds of a random match of his blood to the blood found at the crime scene of 105 to 1.

What are the odds that this suspect was present at the crime scene? What is the probability that this suspect was present at the crime scene?If the odds of a false positive for the DNA profile are one in a thousand, what are the odds that this suspect was present at the crime scene? What is the probability that this suspect was present at the crime scene?

Bayesian Probability• Problem#1

A suspect is seen fleeing the crime. The suspect is positively identified as being at least six feet tall and was wearing a nurse’s uniform. Exactly 5% of the male population is at least 6 feet tall, while 0.5% of the female population is at least 6 feet tall, and 98% of all nurses are female. What are the odds that the suspect is a male.

• Problem#21 million people in America have HIV/AIDS. HIV tests correctly identify a HIV infected person with a positive result 97.7% of the time. HIV tests correctly identify a non-HIV infected person with a negative result 92.6% of the time. If an American gets a positive HIV test result what are the odds that they are infected with HIV? (Assume an american population of 260 million)

• Problem#3Suppose that a barrel contains many small plastic eggs.  Some eggs are painted red and some are painted blue.  40% of the eggs in the bin contain pearls, and 60% contain nothing.   30% of eggs containing pearls are painted blue, and 10% of eggs containing nothing are painted blue.  What is the probability that a blue egg contains a pearl? 

• Problem#4There are 100 people in a room, 20 women and 80 men. 80% of women are blonde, while 30% of the men are blonde. The suspect has blonde hair and is definitely one of the people in the room. What are the odds that the suspect is a female.

• Problem#5The investigator on the case informs you that the odds that the suspect committed the crime are 2 to 1. Your DNA fingerprint analysis of the suspect’s blood gives a 1 in a million probability that it is a random match to the blood found at the crime scene. You also know that your lab has a 1 in a 1000 chance of a false positive. What are the odds that the blood found at the crime scene came from your suspect?