Processing physical evidence discovering, recognizing and examining it;

1. Processing 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.

2. Class characteristics Individual characteristics features that place the item into a specific category features that distinguish one item from another of the same type Lecture: Forensic Evidence and ProbabilityCharacteristics of evidence

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

4. 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?

5. 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 SOLUTION • To 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%

6. 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 cancer P(M | ~C) = 9.6% i.e probability of +ve mammogram given that she does not have cancer The 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.

7. 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 = 80 0.99 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%

8. prior probabilityP(C) . P(~C) The probability that the suspect is or is not guilty prior to presenting this evidence conditional probabilityP(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

9. 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#2 1 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#3 Suppose 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#4 There 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#5 The 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?

10. 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?

11. 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#2 1 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#3 Suppose 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#4 There 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#5 The 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?