WILL THE TEST BE EFFECTIVE IN IDENTIFYING TERRORISTS?
Suppose the U.S. Department of Homeland Security is working on a test
to prevent terrorism: The department has gathered data such as movie rentals,
magazine subscriptions, shopping habits, and the website visits of millions of
Americans. Then, it analyzes the data by a test designed to identify terrorists.
The department is proud of the accuracy of its test: The data from a true
terrorist correctly triggers a positive result 99% of the time. Conversely, the
data from a non-terrorist correctly triggers a negative result 98% of the time.
But, the department estimates that only about 1 in every 10,000
Americans actually is a terrorist. So, you've been called in as an expert statistician to answer this critical question about the test:
If someone tests positive, what's the probability that, in fact, he is a
terrorist?
1) We'll use Hayes's rule; we first make a tree diagram. Fill in the probabilities.
Let T = The person is a terrorist.
Let C = The person is a common non-terrorist.
Let P = The person tests positive-the test result claims he is a terrorist.
tet N = The person tests negative-the test result claims he is a common
non-terrorist.
P
P (T) • P(P I T)
= __
P (T).
=
_
= ---•
= ------
--- •
= -----
-----
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....~) Bayes's rule says:
=
P(B I A)
P(B) • P(A I 8)
P(8) • P(A I B) + P(BC) • P(AI BC)
3) Now, use Bayes's rule to find the probability that someone is a terrorist, given that
he tested positive. In symbols, you are evaluating the probability P(T I P):
P(T , P)
=
5) Fill in the decimal probabilities:
=
4) Fill in the symbols:
P( ). P (
+
)
=
6) Interpret your solution: ~~~~~~~~~~~~~~~~~~~~~
7) Now, start over and find the solution by filling in a 2-way table:
J
Tests
Positive
Tests
Neaative
Terrorists
Common Non-Terrorists
1 000000
8) Justify your reasoning in filling in the two-way table:
9) If someone has tested positive, what's the probability that he is a
terrorist? _~
_
10) Is the test reliable? Explain: _~~~~~~~~~~~~~_
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19
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WILL THE TEST BE EFFECTIVE IN IDENTIFYING TERRORISTS?
Suppose the U.S. Department of Homeland Security is working on a test
to prevent terrorism: The department has gathered data such as movie rentals,
magazine subscriptions, shopping habits, and the website visits of millions of
Americans. Then, it analyzes the data by a test designed to identify terrorists.
The department is proud of the accuracy of its test: The data from a true
terrorist correctly triggers a positive result 99% of the time. Conversely, the
data from a non-terrorist correctly triggers a negative result 98% of the time.
But, the department estimates that only about 1 in every 10,000
Americans actually is a terrorist. So, you've been called in as an expert statistician to answer this critical question about the test:
If someone tests positive, what's the probability that, in fact, he is a
terrorist?
1) We'll use Bayes's rule; we first make a tree diagram. Fill in the probabilities.
Let T = The person is a terrorist.
Let C = The person is a common non-terrorist.
Let P = The person tests positive-the test result claims he is a terrorist.
Let N = The person tests negative-the test result claims he is a common
non-terrorist.
-S"
P (T) • P(P I T)
P
=.0001
P(
-.
q" 9 9
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= ~LO
Key
P(B) • P(AI B)
3) Now, use Bayes's rule to find the probability that someone is a terrorist, given that
he tested positive. In symbols, you are evaluating the probability peT I P):
'
peT I P)
.4) Fill in the symbols:
peT) • P ( PiT)
=
p(T).p(Pl,)
~."
'Ie
P(P\TC;)
+p(\"C).
lO-.$'
5) Fill in the decimal probabilities:
6) Interpret your solution: __
\s
0.49
0.
1
Of"
I_{__
Fe.-
abov+o"2
p
+_~_.s_-t_.5__
~_o_Y"t\_e_o_\')-;--:-~
__
c e'1+
0 ~ "
G~C\"'('C
·h·V~) ~h-er~
be
I'Sq +erl'dl"l,
7) Now, start over and find the solution by filling in a 2-way table:
J
Tests
Positive
-
Terrorists
Tests
Neoative
9q
Common Non-Terrorists
I
.,
l 9.999
91~1402
2O}O9l
91~J 903
100
.
~'CJ,'OO
1 000 000
8) Justify your reasoning in filling in the two-way table:
S\{\C~·
\
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e.ve-r'l . \Ol 00'0
A~en·c.""'J
+e,......o('l~h)
~r-e.
\.v~c~V')
\Y)+c
-rht. 100. T~i~ le~v~s 94', ')00 no\')- +crrcJt'lst~. 11~99
0."':\ +~t \''I/"j'3 ",a \'wJ} ~'Sp.(.+iv'Jt I O~ H. 9 'i'7. ",.J '(13'2., ••.a+-CS.
9) If someone has tested positive, what's the probability that he is a
terrorist?
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10) Is the test reliable?
a..bo ...A- l\'"
Explain:
No) "+-
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