THE PRODUCT RULE
Most of you know this . . .
If you flip a fair coin once, the probability that it will come
up heads is ½ (one out of two).
½ = 50%
1
You plan to flip a fair coin twice. What is the
probability that it will come up heads both times?
96%
1. ¼ or 25%.
2. 1/3 or 33%.
3. I don’t have a clue.
...
0%
Id
on
’t
ha
ve
.
or
33
%
1/
3
¼
or
25
%
.
4%
2
The probability that two independent events will both occur is
equal to the probability that the first event will occur
multiplied by the probability that the second event will occur.
•
Probability of heads on first flip: ½
•
Probability of heads on second flip: ½
• Probability of heads both times: ½ * ½ = ¼
• P (E1 and E2) = P(E1) * P(E2)
4
Seven heads in a row
½ * ½ * ½ * ½ * ½ * ½ * ½ = 1/128 = 0.8%
5
Hypo. Assume that the probability that a car chosen at
random will be a convertible driven by a young man is
1/100. The probability that a car chosen at random will
contain beer is 1/10. The probability that a car chosen at
random will be a convertible containing beer driven by a
young man is 1/100 x 1/10.
72%
1. True
Fa
ls
e
False
Tr
ue
2.
28%
6
When two events are independent, the probability that both
events will occur is equal to the probability that the first event
will occur multiplied by the probability that the second event
will occur.
A different approach is required when the events are not
independent . . .
9
Example – dependent events
Assume:
Probability that a random man will have a beard:
1/5
Probability that a bearded man will have a moustache: 4/5
Then:
Probability that a man chosen at random with have both a
beard and a moustache is:
1/5 * 4/5 = 4/25 = 16%
10
Defendant’s baby died suddenly, with no medical
explanation. Her second baby died the same way. She is
charged with murder. The defense claims that both babies
died a spontaneous “crib death” (SIDS). Assume that SIDS
causes the death of one baby in 2000. The probability that
defendant’s two babies died of SIDS is -[Reference: Sally Clark case and Peter Donnelly on TED.]
69%
1. 1/2000 * 1/2000
2. Unknown
Un
kn
ow
n
1/
20
00
*1
/2
0.
..
31%
11
People v. Collins, p. 872
Supreme Court of California, 1968
Prosecutor’s probabilities:
Partly yellow car
1/10
Man with mustache
1/4
Woman with ponytail
1/10
Woman with blonde hair
1/3
Afr-Am man with beard
1/10
Interracial couple in car
1/1000
1/10*1/4*1/10*1/3*1/10*1/1000 = 1/12,000,000
What’s wrong with saying there’s a 1 in 12 million
chance of innocence?
12
V was murdered in an alley in London. Police do random
tests for the rare type of hair found on the knife that killed
the victim. Only one person in a million would have hair
that matches it. D’s hair matches. So far, no other
evidence has been obtained. It is accurate to say that -67%
33%
ab
ov
e
Ne
ith
of
t
he
he
Bo
th
th
at
t
er
of
th
e
h.
..
...
he
ch
an
ce
th
at
t
Th
e
ch
an
ce
0%
ab
ov
e
0%
Th
e
1. The chance that the
defendant is innocent is
one in a million.
2. The chance that the hair
came from someone
other than the defendant
is one in a million.
3. Both of the above
4. Neither of the above
13
Suppose the probability that a person chosen at random
would match is one in a million.
In a city of 8 million people, you’d expect 8 people to
match.
If there is no evidence other than the match, those people
are all equally likely to be the source of the hair.
14
TRANSPOSITION ERROR
It is an error to “transpose the conditional.” That is, it is an
error to say that the probability of event 1, given that event
2 is true, is equal to the probability of event 2, given that
event 1 is true.
P(E1|E2) ≠ P(E2|E1)
15
EXAMPLE - TRANSPOSITION ERROR
It would be an error to say that the probability that a person
is a lawyer, given that he is a Justice of the Supreme Court,
is the same the probability that a person is a Justice of the
Supreme Court, given that he is a lawyer.
P(L|J) ≠ P(J|L)
16
Going back to the hypo in which D’s hair matches the hair
on the murder knife, and one person in a million has that
type of hair:
The probability that the hair on the knife would match, given
that someone else is the source, is not the same as the
probability that someone else is the source, given that the
hair matches.
P(M|O) ≠ P(O|M)
17
Source probability error (Koehler’s definition):
. Believing that the probability someone else was the source
is the same as the random match probability. (casebook,
p. 954).
“Because only one person in ten has red hair, therefore
the probability that this hair came from someone other
than the defendant is but one in ten.”
Prosecutor’s fallacy (one definition):
Believing that the probability that defendant is innocent is
the same as the probability of finding a match if the
defendant were innocent (i.e., the random match
probability.)
“Only one person in a million has this type of hair,
therefore the probability that defendant is innocent is
one in a million.”
18
People v. Mountain, p. 883
Court of Appeals of New York, 1985
19
In the Mountain case, the expert should be allowed to
testify that –
39%
32%
18%
4%
0%
bl
oo
“O
d
ur
at
th
re
e
su
sc
lts
e.
ar
.
e
Bo
co
th
“T
ns
of
he
...
th
bl
e
oo
ab
d
ov
at
e
th
e
Al
sc
lo
e.
.
ft
No
he
ne
ab
ov
of
e
th
e
ab
ov
e
7%
“T
he
1. “The blood at the scene
matches the defendant’s
blood.”
2. “Our results are
consistent with the blood
having come from the
defendant.”
3. Both of the above
4. “The blood at the scene
came from the
defendant.”
5. All of the above
6. None of the above
20
Time for some guessing. When an earprint expert testifies
“it’s a match,” the intelligent hearer should believe:
84%
.
ea
su
r
ea
su
r
O
n
w
ha
th
e
m
m
w
ha
th
e
O
n
es
,..
..
no
t.
d
pr
in
tc
ou
l
12%
es
...
4%
Th
e
1. The print could not have
come from anyone other
than the defendant
2. On what he measures,
the print is identical, but a
known percentage of the
population would also
match
3. On what he measures,
it’s identical, and we
don’t know what
percentage of the
population would also
match.
21
In the Mountain situation, “it’s a match” would mean
85%
12%
e
w
ha
tw
O
n
O
n
w
ha
tw
e
m
m
ea
su
re
,
ea
su
re
,..
.
no
t..
.
co
ul
d
bl
oo
d
...
4%
Th
e
1. The blood could not have
come from anyone other
than the defendant
2. On what we measure, the
blood is identical, but a
known percentage of the
population would also
match
3. On what we measure, it’s
identical, and we don’t
know what percentage of
the population would also
match.
22
In People v. Mountain, the serology experts knew the
population frequency of blood types, and could state the
probability that a person chosen at random would match.
In many other forensic identification fields, the experts don’t
know the frequency of the identifying features or the
probability of a random match.
-bite mark
-firearms i.d.
-handwriting i.d.
-fiber matches
-microscopic hair matches
-fingerprints
23
Bullet striations
24
Bullet striations under a comparison microscope
Differences aren’t conclusive:
“there may be dissimilar marks due to random contacts at different
places”
“The weapon gets more worn as time goes on.”
Similarities aren’t conclusive:
--they could be class or subclass characteristics
--coincidences do happen
How does the expert know that this weapon can be identified as
the one that fired the bullet to the exclusion of all others in the
world?
Ans. Experience, including viewing photos of coincidental overlap of
features in bullets fired through different firearms. (“We’ve never
seen this many similarities in bullets fired through different
firearms.”) In other words, he doesn’t know for sure.
25
•
Google reference: Itiel Dror
26
Context effect in forensic identification
27
28
29
How could crime labs guard against context effect
in forensic identification?
30
Forensic identification experts usually don’t know the
random match frequency. (DNA is an exception.)
Nonetheless, they testify:
“It’s a match.”
“This match can be made to the exclusion of every other
firearm in the world.” (Testimony in United States v.
Green, D. Mass. 2005.)
“My basis? Police training and 30 years of experience.”
“Error rate? Zero.”
“I’ve never made a mistake.”
“I’ve passed every proficiency test.”
“Practitioner error is possible, but the method has a zero
error rate.”
31
Report of the National Research Council
February 2009
PDF available from the National Academies Press at:
http://www.nap.edu/catalog/12589.html
32
NAS REPORT HIGHLIGHTS
Much forensic testimony do not meet the basic
requirements of good science.
A federal agency should be created for research and
regulation.
33
THE NRC REPORT ON FINGERPRINTS
National Research Council, Strengthening Forensic Science
in the United States: A Path Forward 5-12 to 5-14 (2009)
Conclusion: There is some scientific evidence that
fingerprint ridge patterns are unique to each person and
persist unchanged through a lifetime, but that doesn’t
mean that latent prints from two different people are
sufficiently different so that they can’t be confused.
More research needs to be done characterizing,
quantifying and comparing latent prints.
34
National Research Council, Strengthening Forensic Science
in the United States: A Path Forward 5-12 to 5-14 (2009)
“’At present, fingerprint examiners typically testify in the
language of absolute certainty. . . . [I]n order to pass
scrutiny under Daubert, . . . claims of ‘absolute’ and
‘positive’ identification should be replaced by more
modest claims . . . . ‘“
“Although there is limited information about the reliability of
friction ridge analysis, claims that these analyses have
zero error rates are not scientifically plausible.”
“[T]he examiners can too easily explain a ‘difference’ as an
‘acceptable distortion’ in order to make an identification.”
35
At a Daubert hearing, you’re crossexamining a fingerprint expert who claims to
have a zero error rate. What questions might
you ask?
(Note: You’re probably not going to win at the Daubert
hearing, but you can use it to find out information that will
be useful to you in cross-examining the expert at trial.
Don’t be afraid to ask open-ended or exploratory
questions.)
36
Itiel Dror, Southhampton
“Here are the prints matched by the FBI in the Mayfield
case. If you think they don’t match, tell me why.”
Results
Expert 1
Expert 2
Expert 3
Expert 4
Expert 5
------
No match
No match
No match
Inconclusive
Match
37
Forensic identification evidence:
Possible reforms
•
•
•
•
•
•
Fact screening to prevent context effect
Evidence lineup
More research
Proficiency tests
Exclusion of evidence
Judicial control over assertions permitted in testimony
– Forbid claiming zero error rate
– Forbid pretense of scientific method
– Allow similarities and differences, but not ultimate
issue
38
The end.
Go to show 42b.
39
Global attacks on early use of
DNA evidence
--Not enough data on population subgroups
-- Challenge to independence of gene forms
at different loci
Possible problems with current DNA
evidence
• Contamination during evidence collection
• Sloppy lab work – mix-ups, failure to follow protocols, etc.
– What kind of proficiency testing does the lab do?
• Use of data from wrong population subgroup
• Misinterpretation of mixed-sample results
• Misleading presentation (Cf. Brown v. Farrell, p. 938 (9th Cir. 2008))
–
–
–
–
Prosecutor’s fallacy
Source probability error
Failure to account for relatives
Random match probability doesn’t account for lab error
Question. DNA from a crime scene is tested in a lab. The
lab finds a genetic profile that would be possessed by one
person
in aa million.
1. 1 in
millionThe lab has a 1% false positive error
rate. The probability that a person chosen at random would
2. 1 even
in 500,000
match
if the sample at the crime scene did not come
from
3. that
1 in person
100 is -0%
D
on
’t
m
bi
t
0%
kn
ow
o.
..
10
0
tin
y
A
in
1
0%
in
ill
m
a
in
1
0%
50
0,
00
0
io
n
0%
1
4. A tiny bit more than
one in 100
5. Don’t know
The problem of lab error can be solved by retesting
the sample at other labs.
1. True
2. False
3. It depends
ep
en
d
s
0%
It
d
se
0%
Fa
l
Tr
ue
0%
Blood found on the victim’s shirt matches that of the
accused. If the prosecution wants to put in testimony that
the random match probability is 1 in a billion, it should also
1. required
True to put in an estimate of the lab’s error rate.
be
2. Falsep. 953)
(Koehler,
3. It depends
ep
en
d
s
0%
It
d
se
0%
Fa
l
Tr
ue
0%
. Margaret A. Berger, Laboratory Error Seen
through the Lens of Science and Policy,
p. 955
This article was not assigned this year. It rebuts an
argument by Prof. Koehler. Professor Koehler
argued that prosecutors in DNA cases should
not be allowed to put in evidence of the random
match frequency unless they also can put in
reliable evidence about the lab error rate.
Professor Berger disagreed. The following
slides ask you to make an educated guess about
grounds for disagreement.
An obstacle to calculating an individual lab’s error
rate is
too many
proficiency tests
would be required
2. You can’t tell
whether a lab has
made an error in a
proficiency test
3. Both of the above
1.
Yo
u
to
o
m
an
y
pr
of
ic
ie
nc
ca
y.
n’
..
tt
el
lw
he
th
Bo
e.
..
th
of
th
e
ab
ov
e
0% 0% 0% 0%
A problem with using a pooled error rate, based
on the historical performance of all labs tested, to
predict the performance of a particular lab is--
pr
ov
e.
im
sa
m
Te
ch
ni
qu
es
no
tt
he
ar
e
la
bs
A
ll
0%
ot
h
0%
e
0%
B
1. All labs are not the
same
2. Techniques improve.
3. Both
Opinion poll. If offered by the defendant, evidence
of the pooled error rate of DNA labs should be -1. Admissible
2. Inadmissible
0%
In
ad
m
is
si
bl
e
Ad
m
is
si
bl
e
0%
If you said “admissible,” then why?
ab
...
.
0%
Bo
th
of
t
he
in
at
..
...
0%
Cr
os
sex
am
3.
0%
if
it
is
2.
Even if it is only
approximate, the pooled
error rate could prevent
jurors from being
prejudiced by testimony
that the probability of a
random match is
vanishingly small.
Cross-examination can
expose problems with
using the pooled error
rate.
Both of the above.
Ev
en
1.
• Question A1, p. 961: What is wrong with saying “there
is but one chance in ten that the hair I found at the scene
came from the defendant” ?
• Question A2, pp. 961-62:
What is wrong with saying “the probability is one in seven
million chances that it could be someone other than the
victim?”
Question A-3, 964: Is Victor overestimating the
danger that he has the disease?
0%
0%
se
be
ca
u
Ye
s,
Ye
s.
H
e’
s
do
...
...
0%
th
...
3.
ec
au
se
2.
No, because the test has
only a 1% false positive
error rate.
Yes. He’s done nothing that
would give him syphilis,
therefore the lab’s false
positive error rate must be
more than 1%
Yes, because even if the
false positive rate is only 1%,
with his history he probably
doesn’t have the disease. .
No
,b
1.
Victor tested positive. The test had a 1% false
positive error rate. That doesn’t mean that Victor
only
has
a 1% chance
of being negative because -1.
It is necessary
to consider
ab
...
..
0%
Bo
th
of
t
he
ap
pl
i.
on
e
ne
ce
ss
...
0%
W
he
n
3.
0%
It
is
2.
evidence other than the test
result in estimating
probability that Victor has
syphilis.
When one applies such a
test to a population that has
a very low baserate of the
disease, more than 1% of
those labeled positive by the
test will be false positives.
Both of the above.
Illustration
Suppose that a syphilis test with a 1% false positive error
rate is applied to a population of 1000 that is completely
free of syphilis. 10 of those tested will test positive. All
of the positives will be false positives.
The same principle applies when the baserate is more than
zero. For example, if a population with a 1% incidence
of syphilis was given a test with a 1% error rate, than
approximately half of those who tested positive would be
false positives.
Question 4, p. 962.
Yes.
No.
Don’t know.
0%
kn
ow
o.
.
0%
D
on
’t
N
0%
Ye
s.
1.
2.
3.
Is Ernie wrong?
Your opponent argues the
“Prosecutor’s fallacy.”
He says that because there is a 1 in 100 chance of a
random match, therefore the chances are 1 in 100 that
your client is innocent.
What’s your answer?
Your opponent argues the defense
counsel’s fallacy.
In a case where there is evidence of guilt other than the
genetic match, he argues that because there are 10,000
other men in the area with the same genetic markers as
his client, therefore there is a one in 10,000 chance that
his client is guilty.
What’s your counterargument?
The end.
The end
61